I had this recently with a relative. I bathed them. She has dementia and hates anything outside her routine so I had her impression of the Churchill dog saying "no no no no no no no" as is her mantra but when asked if she would have let her children refuse to wash she said that she would have forced them. I did NOT force her, I was gentle, firm and encouraging. I also had a reward for after (tea and cake)
Elderly people feel the cold very acutely. I used this as my guide. I ensured that the room was warm. There were heaps of towels and I wrapped a towel around her shoulders and got her safe and comfortable before helping her to remove her footwear. The first time, she washed with the towel around her shoulders until she was relaxed
I talked with her constantly (think loud parenting - well done, you're doing brilliantly, that's it, are you nice and warm, fabulous,)
I encouraged her in and stayed touching her throughout. Reassuring and being a presence.
DH's carers cannot force her to wash. You cannot force her to wash.
If she is fully "with it" you can have a brutally honest conversation with her about it. Give reasons. Listen to her but ensure that the TV, radio etc is off and that she listens to you.
Give a reward. Are there grandchildren/great grandchildren that can visit but might be avoiding because she smells?
If she is not fully "with it" then treat her like your child. Be gentle, firm but fair. Charts and stickers and warnings when something is coming can help.
How's her skin? Dry? Does she have any sores? Any reason you can find for her needing to shower?/avoiding?
Do not force her. You'll only do it once.
You have some unmumsnetty hugs from me. I know how hard it can be xxx