What's causing my thumb tremor after riding? After asking this beginner's question, I took people's advice and started test driving bicycles. I rode for about twenty minutes, so not too long. About fifteen minutes later, I found that my right thumb had developed a serious uncontrollable tremor. Whether I clenched the thumb or relaxed the thumb, it kept shaking. The left was not shaking.

I test drove two bikes (both relatively upright): one where the hand grips + brake handles fit well (though I didn't have padded gloves so that caused some pain) and the second where the brake handles were far enough from the hand grips that I could only reach them with my finger tips. I returned quickly with that bike because it hurt and I didn't feel safe being unable to control the brakes well.

I had ulnar nerve problems with that hand/wrist twenty-odd years ago, though the ulnar nerve numbness ran from my pinkie and ring fingers up to my neck and didn't involve my thumb at all.

Is the thumb tremor simply a result of bad-configuration-for-me handles? Or is it a symptom that I haven't ridden in forever? Was it the lack of proper padded gloves? All three?
The lack of padding on your gloves is unlikely to have been the problem in such a short ride. The lack of grip in adverse weather conditions is more likely to cause it. You mentioned that you were holding the bars very tight. With typical stock handlebar grips it's perfectly possible to apply a lot of pressure on the nerves at the base of your thumb, and this can cause numbness and loss of control.

The good news is it should get better fast. When I've had this issue (often sweaty hands and unexpectedly rough terrain) it's faded within a couple of hours. Others report it taking a bit longer. 

There are several things you can do:


In warm/dry conditions bare hands have decent grip.
Bike gloves don't have to be expensive. My good gloves were under £/€/$10 on eBay. 
Even minor changes to your hand position can relieve the pressure. 
Riding one handed allows you to flex and relieve the other hand. It might seem unrealistic when you've just started riding but it's a useful skill, essential in traffic for signalling where you're going. Practicing riding with one hand while the other hovers near the bars is a good start and will help your hand. 
Ergonomic grips spread the pressure, and provide more hand positions on flat bars. I've used the Ergon GP2 and GP3 with bar-ends on my hybrid, and they're good for riding all day. 
Adjusting the bike to fit (especially so you're not using your hands to stop yourself sliding forwards all the time) will also help. This is something you can do yourself or with the help of a friend, but you'll need some longer rides to test it. 


Finally if it doesn't get better within a day or so this time, or keeps coming back, you may need to get it checked out