Great advice, as usual. After all, if all the clock needed was a bit of attention and a dash of lubricant, maybe the speedometer is the same! So today I pulled it out and started cleaning and looking - though I didn't have enough time to get all the way into the mechanism, I've got a pretty good idea of how it all goes together thanks to Larry's photos and my investigations.
I think I'll replace all the lightbulbs while I've got it out, too.
And guess what? The original owner's meticulous records come in handy here! One of the business cards she had - one of the few shops that's still around - is a speedometer/gauge repair shop, that specializes in classic mechanical gauges. So if I do need to take it in somewhere, I know just where to go.