Stinging nettle is apparently used since ancient times, also for the problems or diseases indicated in the question:
Some popular medicinal plants and diseases of the Upper Palaeolithic in Western Georgia. (2015)
U. dioica is considered a good remedy for rheumatism, asthma, anaemia and gout (Launert, 1981; Foster and Duke, 1990; Adams et al., 2009; Hayta et al. 2014; Zlatović et al., 2014). Moreover, nettles are used to treat burns, viral, bacterial and fungal diseases (Foster and Duke, 1990; Adams et al., 2009). Nettle is also a good pain reliever for rheumatism (Randall et al., 1999; Bown, 1995; Hajhashemi and Klooshani, 2012). New investigations showed that these recent results confirm the folkloric use of the plant extract to treat painful and inflammatory conditions. Further studies are needed to characterise the active constituents and the mechanism of action of the plant extract (Hajhashemi and Klooshani, 2012).
It should be noted that as phrased in the question: "taking a bath in stinging nettles" – it is probably a bit of a misconception. It is not really needed to use fresh, and still stinging, nettles. In saunas broom sticks are first dried and then dipped into hot water before using them, reducing the sting substantially. It seems to confer beneficial effects when used as a completely non-stinging extract in a bath as well.
The contents of the burn hairs might do something on their own and help the other juices enter the skin but most of the more interesting contents (like cytokine modulator hydroxyoactedecatrienic acid) are not primary constituents in in the burn hairs. It gets further complicated by newer findings calling into question the whole understanding of the mechanism of action as well as the constituents of the defensive trichomes of nettles.
Pharmacognostical review of Urtica dioica L. (2014) The plant methanolic extract at doses 200 and 400 mg/kg has been shown to inhibit dose dependently acetic acid‐induced abdominal twitches and carrageenan induced paw edema. The N‐Methyl‐D‐aspartate (NMDA) injection‐induced brain lesion and subsequent inflammation in wistar rats significantly decreasing the nuclear factor kappa B (NF‐kB) binding activity to DNA on administration of Urtica dioica leaf supplementation which suggests a signi cant anti‐inflammatory effect. […]
The compounds quercetin‐3‐O‐rutinoside, kaempherol‐3‐O‐rutinoside and isorhamnetin‐3‐O‐glucoside present in the methanolic extract of the aerial parts of the plant contribute to the immunomodulatory activity of the plant.
Trying to explain another possible mechanism of action is:
Scientific Basis of Botanical Medicine as Alternative Remedies for Rheumatoid Arthritis (2013)
[…] and Urtica dioica. Scientific research has demonstrated that these herbs have strong anti-inflammatory and anti-arthritic effects. A wide range of phytochemicals including phenolic acids, phenylpropanoid ester, triterpene glycosides, phthalide, flavonoids, triterpenoid saponin, diterpene and triterpene have been isolated and demonstrated to be responsible for the biological effects of the herbs. Understanding the mechanisms of action of the herbs may provide new treatment opportunities for RA patients.
One such RA medication recently approved is a pill, not a topical ointment, containing mostly lipophilic chemicals of the plant extract. Of thes ingredients 13-Hydroxyoactedecatrienic acid, which was proven to inhibit pro-inflammatory cytokines IL-1β and TNF-α, thereby directly interfering with the disease process, among other modes.
While it seems quite true that:
The unsatisfactory outcome of current drugs for treating rheumatoid arthritis has led to the consideration of alternative medicine. Modern biological immunomodulators, while showing efficacy in the treatment of RA, may carry as yet unknown side effects. Botanical drugs from traditional Chinese medicine have been used to treat rheumatoid arthritis since ancient times. In recent years, advanced and sophisticated research has been done on many medicinal herbs and some have demonstrated possible efficacy. With the help of modern science and technology, the chemical constituents and underlying mechanisms are being developed.
Reviewing the application of herbal medicine and its effectiveness:
Anti-Inflammatory Drugs and Herbs with Special Emphasis on Herbal Medicines for Countering Inflammatory Diseases and Disorders - A Review. (2018, in print)
Conclusion: Natural herbs are safe, effective and better options as anti-inflammatory agents than synthetic ones. The phytoconstituents are as effective with the comparable mechanism of action as synthetic molecules.
The chemical constituents are there to explain quite a bit of action, also applied topically. As the paragraph cited above indicates, until recently systematic research into this avenue was not really attractive for pharmaceutical companies developing drugs. Only the persistent failure of conventional new drug development to arrive at a satisfactory relationship between effects and side-effects has renewed scientific interest in the herbal medicine direction.
That means there is a comparative lack of research comparing thousands of years of folklore uses and believe in "it works" with decades of pharmacological research into synthetic drugs and their result of "it works almost OK, but not great and has a number of side effects". The current verdict on reviewing the evidence systematically still stands:
Topical herbal therapies for treating osteoarthritis(2013)
We are also uncertain if other topical herbal products (Marhame-Mafasel compress, stinging nettle leaf) improve osteoarthritis symptoms due to the very low quality evidence from single trials. No serious side effects were reported.
Although the mechanism of action of the topical medicinal plant products provides a rationale basis for their use in the treatment of osteoarthritis, the quality and quantity of current research studies of effectiveness are insufficient. […] Further high quality, fully powered studies are required to confirm the trends of effectiveness identifed in studies so far.
Depending on target and mode of application: even a bath in still stinging nettles might help with joint pains. That may be a bit of an unpleasant tradition that does probably represent the oldest and cheapest but not even the most effective route to deliver the drugs contained in nettle. Stinging nettle has many uses and a wide range of pharmacologically active substances. But high quality research on the whole range of effects for stinging nettle and all its potential targets and applications is still not available in sufficient quantity to reach a definitive conclusion.