Not sure what would be good evidence (along strict Skeptics SE lines) for the main (forecasting) question (since the data question is merely a prop). Perhaps that other forecasters came up with similar numbers?
Starting in 2030, (Bloomberg New Energy Finance) BNEF predicts that 26 million EVs will be sold annually, representing 28 percent of the world's new cars sold.
Their report is here. Frankly BNEF seemed more worried if there would be enough electricity for them than if there would be enough raw materials... So maybe they've ignored that latter angle, not sure.
The IEA seems even more optimistic:
In 2030, in the New Policies Scenario, which includes the impact of announced policy ambitions, global electric car sales reach 23 million and the stock exceeds 130 million vehicles (excluding two/three-wheelers). In the EV30@30 Scenario, which accounts for the pledges of the EVI EV30@30 Campaign to reach 30% market share for electric vehicles (EVs) by 2030 (excluding two/three-wheelers), EV sales reach 43 million and the stock is more than 250 million.
Both of these studies seem to see the price of components (in particular batteries) as a major factor affecting the forecast. The latter (IEA) study does touch a bit more on raw material demands, but it starts that discussion by factorig in projected technological changes in battery chemistry:
It is expected that by 2025 batteries will increasingly use cathode chemistries that are less dependent on cobalt, such as NMC 8111, NMC 622 or NMC 532 cathodes in the NMC family or advanced NCA batteries. This will lead to an increase in energy density and a decrease of battery costs, in combination with other developments (e.g. the availability of silicon-graphite chemistries for anode technology). [...]
Increasing electric mobility and the ramp-up of related battery production imply increased larger demand for new materials in the automotive sector. The type of materials will vary according to advances in battery chemistry technologies. Assuming a mix of battery chemistry categories of 10% NCA, 40% NMC 622 and 50% NMC 811 for 2030, in the New Policies Scenario, the demand for cobalt increases to about 170 kilotonnes per year (kt/year), lithium demand to around 155 kt/year, manganese to 155 kt/year and class I nickel (>99% nickel content) to 850 kt/year. In the EV30@30 Scenario, the larger scale uptake of EVs implies volumes in 2030 more than twice as high as in the New Policies Scenario. For cobalt and lithium, these volumes mean that demand in the New Policies Scenario exceeds current supply. For class I nickel, this is the case in the EV30@30 Scenario. Cathode battery chemistry significantly affects the sensitivity of the demand of metals, particularly cobalt.
Cobalt and lithium demand are expected to significantly rise in the period to 2030. Cobalt demand has the largest variation due to the type of cathode chemistry. Cobalt and lithium supplies need to scale up to enable the projected EV uptake.

Notes: NPS = New Policies Scenario, kt = kilotonnes.
The IEA doesn't seem to discuss raw material limitation much beyond that, i.e. whether the projected increases in mining/production of raw materials is reasonable, but I guess they assume it is since they come up with an estimate that implies such an increase in raw materials availability...
There are actually a few more paper on the topic of raw materials required/available, some a bit less optimistic than others e.g. Steubing et al. (2020) in Nat. Comm. Materials:
We find that in a lithium nickel cobalt manganese oxide dominated battery scenario, demand is estimated to increase by factors of 18–20 for lithium, 17–19 for cobalt, 28–31 for nickel, and 15–20 for most other materials from 2020 to 2050, requiring a drastic expansion of lithium, cobalt, and nickel supply chains and likely additional resource discovery. However, uncertainties are large. Key factors are the development of the electric vehicles fleet and battery capacity requirements per vehicle. If other battery chemistries were used at large scale, e.g. lithium iron phosphate or novel lithium-sulphur or lithium-air batteries, the demand for cobalt and nickel would be substantially smaller. Closed-loop recycling plays a minor, but increasingly important role for reducing primary material demand until 2050, however, advances in recycling are necessary to economically recover battery-grade materials from end-of-life batteries. Second-use of electric vehicles batteries further delays recycling potentials. [...]
Due to the fast growth of the EV market, concerns over the sustainable supply of battery materials have been voiced. These include supply risks due to high geopolitical concentrations of cobalt and social and environmental impacts associated with mining as well as the availability of cobalt and lithium reserves and the required rapid upscaling of supply chains to meet expected demand.
And just for that last para they cite several other papers:
Olivetti, E. A., Ceder, G., Gaustad, G. G. & Fu, X. Lithium-ion battery supply chain considerations: analysis of potential bottlenecks in critical metals. Joule 1, 229–243 (2017).
Van den Brink, S., Kleijn, R., Sprecher, B. & Tukker, A. Identifying supply risks by mapping the cobalt supply chain. Resour. Conserv. Recycl. 156, 104743 (2020).
Banza Lubaba Nkulu, C. et al. Sustainability of artisanal mining of cobalt in DR Congo. Nat. Sustain. 1, 495–504 (2018).
Thies, C., Kieckhäfer, K., Spengler, T. S. & Sodhi, M. S. Assessment of social sustainability hotspots in the supply chain of lithium-ion batteries. Procedia CIRP 80, 292–297 (2019).
Weil, M., Ziemann, S. & Peters, J. The Issue of Metal Resources in Li-Ion Batteries for Electric Vehicles. in Behaviour of Lithium-Ion Batteries in Electric Vehicles: Battery Health, Performance, Safety, and Cost (eds Pistoia, G. & Liaw, B.) 59–74 (Springer, 2018).
They comment on the last study in that list
Weil et al. assess the material demand for EV batteries at the global level and find that shortages for key materials, such as Li and Co, can be expected. However, their model does not investigate the influence of battery chemistry developments (e.g., improved NCM chemistries or novel lithium-sulphur (Li-S) and lithium-air batteries (Li-Air)) as well as alternative fleet and different recycling scenarios.
Here, we go beyond previous studies by developing comprehensive global scenarios for the development of the EV fleet, battery technology (including potentially game changing chemistries such as Li-S and Li-Air) as well as recycling and second-use of EV batteries. We assess the global material demand for light-duty EV batteries for Li, Ni, and Co, as well as for manganese (Mn), aluminum (Al), copper (Cu), graphite, and silicon (Si) (for model details, see Supplementary Fig. 1). We also relate material demands to current production capacities and known reserves and discuss key factors for reducing material requirements. The results presented are intended to inform the ongoing discussion on the transition to electric vehicles by providing a better understanding of future battery material demand and the key factors driving it.
This paper actually relies on (some) IEA estimates as well for potential EV market growth in order to gauge potential demand, which seems to include the aforementioned EV30@30 target (actual they cite an updated IEA 2020 report for that):
We base our scenarios on two scenarios of the International Energy Agency (IEA) until 2030: the Stated Policies (STEP) scenario, which incorporates existing government policies and the Sustainable Development (SD) scenario, which is compatible with the climate goals of the Paris agreement and includes also the target of reaching a 30% global sales share for EVs by 20303. According to these scenarios, EVs will make up 8–14% of the total light-duty vehicle fleet by 2030, of which 89–166 million are battery electric vehicles (BEVs) and 46–71 million are plug-in hybrid electric vehicles (PHEVs).
And to get the estimates for required battery material, one has to work back from their (estimated) energy density and the estimated demand for energy storage for EV. I'll spare you most the details since the paper is open access, but e.g. it does get a bit complicated if one is to discuss projected tech changes:
The most likely NCX scenario follows the current trend of a widespread use of lithium nickel cobalt aluminum (NCA) and lithium nickel cobalt manganese (NCM) batteries (henceforth called the NCX scenario with X representing either Al or Mn)17. Battery producers are seeking to replace costly cobalt with nickel, which has led to an evolution from NCM111 to NCM523, NCM622, and NCM811 batteries (numbers denote ratios of nickel, cobalt, and manganese)17 and NCM955 (90% nickel, 5% cobalt, 5% manganese) are expected to be available by 2030. [...]
The LFP scenario considers the possibility that LFP (LiFePO4) batteries will be increasingly used for EVs in the future. The principle drawback of LFPs is their lower specific energy compared to NCA and NCM chemistries, which negatively impacts fuel economy and range of EVs. Advantages of LFPs are lower production costs due to the abundance of precursor materials, safety due to better thermal stability, and longer cycle life20. While LFP batteries have seen their main application in commercial vehicles, such as buses, there are prospects of a more widespread use of LFPs in light-duty EVs (e.g. Tesla has recently announced to equip the Chinese version of its Model 3 with LFP batteries).
The Li-S/Air scenario assumes these experimental techs become available by 2030. And below are the material estimates relative to current production/reserves.


As for the authors' conclusion(s), which also compare their materials estimates with prior studies focused on that:
Given the magnitude of the battery material demand growth across all scenarios, global production capacity for Li, Co, and Ni (black lines in Fig. 3) will have to increase drastically (see Supplementary Tables 9 and 10). For Li and Co, demand could outgrow current production capacities even before 2025. For Ni, the situation appears to be less dramatic, although by 2040 EV batteries alone could consume as much as the global primary Ni production in 2019. Other battery materials could be supplied without exceeding existing production capacities (Supplementary Tables 9 and 10), although supplies may still have to increase to meet demands from other sectors. The known reserves for Li, Ni, and Co (black lines in Fig. 4) could be depleted before 2050 in the SD scenario and for Co also in the STEP scenario. For all other materials known reserves exceed demand from EV batteries until 2050 (Supplementary Table 5). In 2019 around 64% of natural graphite and 64% of Si are produced in China, which could create vulnerabilities to supply reliability. However, synthetic graphite has begun to dominate the LIB graphite anode market (56% market share in 2018) due to its superior performance and decreasing cost over natural graphite. Thus, among EV battery materials Co and Li, and to a lesser extent Ni and graphite, can be considered to be most critical concerning the upscaling of production capacities (see Supplementary Table 9), reserves and other supply risks, which confirms previous findings [five papers cited here] even without taking into consideration the potential additional demand from heavy-duty vehicles and other sectors. In contrast to Li and Ni, Co reserves are also geographically more concentrated and partly in conflict areas, thus increasing potential supply risks. Battery manufacturers are already seeking to decrease their reliance on cobalt, e.g., by lowering the Co content of NCM batteries; however, as shown in Fig. 3, an absolute decoupling is unlikely to occur in the coming decades. Shortages could also occur at a regional level, such as the access to Li and Ni for Europe. Obviously, it is possible that the outlined supply risks change, e.g., with the discovery of new reserves.
According to our model, lithium demand for EV batteries in 2050 (0.6–1.5 Mt) could be significantly lower than projected by Weil et al. (1.1–1.7 Mt) and likely higher than projected by Hao et al. (0.65 Mt), Deetman et al. (0.05–0.8 Mt), and Ziemann et al. (0.37–1.43 Mt). For cobalt our estimations (0.25–1.25 Mt) are in-line with the predictions by Weil et al. (0.3–1.1 Mt) despite important differences in underlying scenarios and likely considerably higher than Deetman et al. (0.06–0.62 Mt). For nickel our estimations (1.5–7.6 Mt) partly overlap but are generally higher than those by Weil et al. (0.6–2.6). There are thus notable uncertainties concerning the primary material demand for EV materials related to several key factors that could be strategically addressed to mitigate supply risks. Probably the most important factor is the future required battery capacity. A sensitivity analysis is shown in Fig. 4 for two extreme battery capacity cases, i.e., if all EVs were PHEVs with small 10 kWh batteries or if all EVs were large SUVs with 110 kWh batteries, such as Tesla Model S Long Range Plus. While it is unlikely that the global average EV battery capacity will be close to either end of this range, this analysis illustrates the high importance of this factor. The demand for battery capacity depends on technical factors, such as vehicle design, vehicle weight, and fuel efficiency, and perhaps even more importantly, on socio-economic factors, such as the future EV fleet size (see also Fig. 4), consumer choices concerning the size and ranges of EVs, the cost of EV batteries and raw materials, the development of alternative transportation means and technologies (e.g., fuel cell EVs), and policy.
They also discuss recyling to some extent, but note that it doesn't look like an immediate opportunity
Truly circular EV batteries will not be available anytime soon. Over the next decades we first need to produce the EV battery stock for a large fleet, mostly from primary materials. [...] The most economically and environmentally promising technology for closed-loop recycling, although currently largely unproven outside of the lab, is direct recycling, which could recover cathode material “as is” without intermediate smelting or leaching step.
And from the (50 pages!) of supplementary materials for this paper, the most relevant tables mentioned above:


As you can see, these table (unlike earlier graphs) alas discuss the materials targets mostly with respect to 2050 rather than 2030 (as the OP/claim wants), but you have to deal with what you can find (published) when it comes to complex forecasts like this...
The EU also acknowledged the concern, but only with respect to cobalt (as far as I could find), in this (somewhat cheesy) infographic.

Road transport is undergoing a radical transformation with
the switch from conventional to electric vehicles (EVs): as
many as 130 million electric cars are expected to circulate
worldwide in 2030, compared to 3.2 million in 2017. As a
result, the worldwide demand for cobalt (a crucial element for
the most common types of lithium-ion batteries used in EVs)
could potentially increase threefold within the next decade,
even assuming the future adoption of low-cobalt chemistries
in EV-battery manufacturing.
While supply will be meeting demand until 2025, projections
show shortages beyond this point in time. Cobalt prices have
already tripled between 2016 and 2018 and, since they
account for a significant part of the battery production costs,
a further escalation might also impact EV prices. Substituting
cobalt with other metals is technically possible and could
reduce the 2030 EV market demand by 29 %; however, this
will not be enough to fill the demand-supply gap alone.

That EU brochure is based on a longer, 104-page report. The supply-demand balance is covered on pages 54-75 with various tech-change scenarios. Even in the best-case (demand) scenario where substantial recycling occurs, they still predict a shortfall by 2030.
