Yes. With ample caveats.
It shows antiviral activity in the lab, it is often used traditionally in virus-induced diseases. But as an actual prime frontline medicine against viral infection clinical evidence for its use it lacking.
Its antiviral activity is well established in vitro. But apart from laboratory settings its efficacy as an actually therapeutic agent was classified by Kommission E as an established traditional medication, but only for catarrhs of the upper respiratory tract, bronchitis and pertussis (whooping cough)'. All other, including viral infection targets, were summarily dismissed as "not enough evidence", despite ample evidence for its traditional use in a far wider applications. (Monograph 00325, BAnz. No.228/05.12.1984, BAnz. No.226/02.12.1992, BAnz. No. 50/13.03.1990. Transferred to: 12 November 2013, EMA/HMPC/342332/2013, Committee on Herbal Medicinal Products (HMPC): "Community herbal monograph on Thymus vulgaris L. and Thymus zygis L., herba", PDF). For common cold, influenza etc it is still useful as an expectorant and adjuvant. Not in the least as its anitbacterial activity is far better documented. Its aroma therapy value for similar conditions is equally well documented for use, but also lacking evidence of usefulness, simply for the lack of clinical studies.
That non-cytotoxic preparations of this herb display antiviral activities in human cell lines is evidenced in multiple studies. For example:
— Akram Astani, Jürgen Reichling & Paul Schnitzler: "Comparative Study on the Antiviral Activity of Selected Monoterpenes Derived from Essential Oils", Phytotherapy Research 24: 673–679 (2010) DOI: 10.1002/ptr.2955.
— Jürgen Reichling, Paul Schnitzler, Ulrike Suschke & Reinhard Saller: "Essential Oils of Aromatic Plants with Antibacterial, Antifungal, Antiviral, and Cytotoxic Properties – an Overview", Forschende Komplementärmedizin, 2009;16:79–90. DOI: 10.1159/000207196
An inherent problem for this essential oil product is that in the vast family of thyme species even just the one Thymus vulgaris produces a wide ranging spectrum of compounds. In its thymol fraction 1,8-cineol for example is frequent in Spanish specimen, but entirely absent in French specimen.
Another one is that frequently the volatile compounds in liquid phase inhibit or even 'kill' viruses in a petri dish but also exhibit cytotoxic activity detrimental for the host's own cells.
For systemic application this might be a problem of bioavailability and delivery. But in topical applications this is less of a problem.
— Koch C, Reichling J, Schneele J & Schnitzler P.: "Inhibitory effect of essential oils against herpes simplex virus type 2", Phytomedicine. 2008 Jan;15(1-2):71-8.
The most interesting research in that regard is that essential oil vapors were shown to be effective in such low concentrations as to be benign towards epithel cells over a short but 'long enough' timespan:
[…] Thymus vulgaris displayed 100% inhibitory activity at 3.1 μL/mL concentration. Under these conditions the vapors showed no measurable adverse effect on epithelial cell monolayers. This suggests that these oils in their vapor phases could be potentially useful in influenza therapy. The oil vapors were also evaluated for possible direct effects on the principal external proteins of the influenza virus, namely the HA (hemagglutinin) and NA (Neuraminidase). Several of the vapors inhibited the HA activity, but not the NA activity, suggesting that interaction with HA is a possible mechanism for the antiviral activity. Thus some of these oil vapors could have therapeutic benefits for people suffering from influenza, and possibly other membrane containing respiratory viruses.
— Selvarani Vimalanathan & James Hudson: "Anti-influenza virus activity of essential oils and vapors", American Journal of Essential Oils and Natural Products 2014; 2 (1): 47–53. (PDF)
Again, any conformation for efficacy in a clinical setting is lacking for this.
For more perspective on the general outlook of 'herbal essential oils againstas anti-virals', as even "peppermint is showing some antiviral activity":
— Katarzyna Wińska et al.: "Essential Oils as Antimicrobial Agents—Myth or Real Alternative?", Molecules. 2019 Jun; 24(11): 2130. doi: 10.3390/molecules24112130, PMCID: PMC6612361, PMID: 31195752
The textbook summary is thus:
Thyme oil demonstrates antiviral activity against HSV-1, HSV-2 and an aciclovir-resistant strain of the virus (Koch et al 2007, Nolkemper et al 2006, Schnitzler et al 2007). One study found that the oil decreased plaque formation by more than 90% when preincubated with HSV-2; however, no effect was observed when the oil was added prior to infection or after the absorption stage (Koch et al 2007). It was suggested that thyme essential oil interferes with the viral envelope.
Thyme has not been significantly investigated in controlled studies, therefore information is generally derived from evidence of activity and traditional use.
What will this herb do for me?
When taken internally, thyme is used to treat bronchitis, symptoms of the common cold, diarrhoea and dyspepsia. It is also used as an antiseptic gargle for sore throats and can be diluted and applied externally to minor wounds.
Thyme has strong antibacterial activity and antifungal actions and is used as a treatment for common infections such as bronchitis and upper respiratory tract infections. It is also used as a food preservative, for relieving symptoms of dyspepsia and diarrhoea.
— Lesley Braun & Marc Cohen (eds): "Herbs & Natural Supplements. An evidence-based guide.", Churchill Livingstone Elsevier: Chatswood, 42015.