found a study
A very weak study in this context, for multiple reasons: Somewhat obscure journal (impact factor of 1.06), end point was the swimmers' mood, which may also influence pain reporting, intervention was winter swimming, which is much dfifferent from cold showers in many ways (shower != sports activity), the pool of participants was structurally non-overlapping (children versus people who go winter swimming!), the pain relief is confounded by also feeling "more energetic, active and brisk", compared to controls who did not do that sports activity, the list goes on.
A plausible explanation of your doctor's observation is that cold showers made people who already had rheumatism more likely to report it to doctors.
Yea.
Thus spake Eliezer:
It seems that many here might have outlandish ideas for ways of improving our lives. For instance, a recent post advocated installing really bright lights as a way to boost alertness and productivity. We should not adopt such hacks into our dogma until we're pretty sure they work; however, one way of knowing whether a crazy idea works is to try implementing it, and you may have more ideas than you're planning to implement.
So: please post all such lifehack ideas! Even if you haven't tried them, even if they seem unlikely to work. Post them separately, unless some other way would be more appropriate. If you've tried some idea and it hasn't worked, it would be useful to post that too.