There are two different issues here:
1. Yes, a drive with a 32-bit installation of Windows will do just
fine with a 64-but motherboard and CPU.
2. But trying to use an existing Windows installation on a drive when
the motherboard has changed (whether or not the “bitness” has
changed), may or may not work. At the least, you normally have to do a
repair installation. But sometimes that’s not enough and a clean
reinstallation is necessary.