What do I do when mom won't bathe?

I don't know if this will help, but I have read that resistance to bathing is not to stay dirty. It's from fear of being cold, the echo and noise of bathing, and vulnerability. Not knowing your ability to do so, I can suggest old fashioned sponge baths, so she doesn't have to be completely naked all at once. Medicated powder is a great barrier and relief for the rashes. Dry shampoo spray to ride out the time between real shampoos. There are a lot on the market these days for ladies to use on their hair extensions, so there are plenty of choices at all prices. I guess if you have to pick your battles, maybe prioritize what part of her gets cleaned in a day simply to stay on top of the rashes and use powders & deodorant sprays, or cornstarch to control B.O.
Face/hands, pits, privates, & feet are the essentials really.

When you can get in-home assistance or move her to a facility with roll-in showers, it will be easier for both of you.