"About 12 gauge" is hardly an engineering statement. Didn't they use a micrometer?
Possibly 2mm diameter.
(maybe slightly
#12 wire is 2.053mm dia. #10 is 2.588.
Best practice would be to wind for the same number of amp-turns as before.
From wire tables, changing from #12 to #10 wire,the resistance of the same number of turns will decrease by a factor of approximately 0.6. The current will correspondingly increase by a factor of approximately 1.6, hence so will the amp-turns, assuming the supply voltage stays constant. How much the strength of the magnet will increase depends upon the magnetic properties of the magnet iron. If the magnet was originally operated close to saturation, the increase in pulling strength may not be very much.
The power dissipated in the copper is proportional to current squared, hence 1.6 for the same voltage. The coil will run hotter.
There's also the issue of supply regulation. Will the transformer supply the increased current without significant voltage drop?
Why did they rewind it using #10? Didn't they have any #12? I'm surprised they didn't have metric sizes, there are a lot of Japanese and European motors around.