Looking in the Corpus of Global Web-Based English, I found more of both in American and more of shift in both dialects.
Using a web-based corpus is possibly a bit funny for this, since authorship isn't known and they might be writing for an international audience (among other reasons). So, what happens when we look at books published in US and UK? I checked out Google Books--which also has a lot of problems in classifying data, but we hope that the sheer amount of well-classified data limits the effect of the poorly classified examples. (E.g. I once found that because a publisher put its founding date in the 18th century on its title pages, Google books thought that its books were written in the 18th century. Including the ones about television.)
The books data seems to explain things better. (NB: Firefox doesn't seem to be able to handle the dates on the bottom line, but other browsers can. But if you're on Firefox, scrolling over the chart should show dates. Or maybe this is just my Firefox.).
Here's the American:
What we have here is that both shift and change are earlier in American than British (though the first change gears are pretty close to one another--so that's just a matter of new technology needing new expressions). Then shift was introduced in the US in the 1910s, and fairly steadily rose until it overtook change in the early 1960s. The Americanness of the introduction is confirmed in the OED where all examples for its first several decades are American--though the OED does not label it as an AmE (or 'orig. U.S.') usage. When shift got to be used more than change in AmE, it started to be really noticed in BrE and now we have a situation where both dialects tend to use the newer verb shift, but haven't forgotten the older one--though change is still more common in BrE than AmE.
Now back to marking!
Postscript 3 March: I promised ages ago in the comments that I'd address the number of gear(s)--something that the original post should have done! So here are the numbers from the Global Web-Based English corpus.
For AmE, gears is definitely the winner, no matter the verb.
But in British English, the older change goes with singular gear more, and the more American shift goes with the more American plural gears.