If climbing shoes are resoled, will they fit differently afterward? If so, in what way does it change (narrower, shorter, tighter, looser, etc..)?
Not unless they did a really bad job.
Shoe size starts with the sole. It would be really hard to even sew in an upper to a wrong sized sole. If you told the cobbler you want less width they could stitch in a bit but not very much.
It will take a bit of time to break the sole in but not nearly as long as the original break in period.