I'm wondering how you're getting on with this thorny problem. I've made some progress with my mother and thought I'd post an update.
The shower was installed and sealed just before Good Friday. The sealant had to be left a good 24 hours to dry properly, then my sister arrived for a visit and was the first one to use the new shower. She was very impressed with it and was encouraging Mum to use it but Mum was resisting.
I cunningly extracted a promise from Mum that the next day, after my sister had returned home, she would have a shower. The next day arrived, but Mum was resisting with every excuse she could think of. She hadn't had a bath in months and hadn't washed her hair for about 3 weeks. I took her into the bathroom and showed her the controls and how gentle the warm water was. I showed her the stool and the grab rail and assured her I would be outside the door whenever she needed me. Still she refused. I took her into the bedroom and we sat on the bed. I cajoled and tried every argument I could think of to persuade her. I said it was a health issue and I could not let it go on any longer, then said that I would not leave until she'd had a shower. She said "You have to go to work this evening." I said, "I'll ring them and explain my elderly parent is ill. I have all the time and patience in the world. I will wait 4, 6, 8, 10 hours or HOWEVER LONG IT TAKES until you cave in, so please just co-operate." She was shocked at this tactic. I then got a vest out of her drawer and asked her to take her top off and put that on for some privacy. She said, "OK, I'll change my underwear but I'm not going in the shower". I said - "we'll see". Once the vest was on (which was long enough to cover her nether regions too), I reached underneath and unhooked her bra...that was when a large pile of dead flakey skin fell out of it onto her lap.
She was really shocked and said "What's that?!?" I explained it was her dead skin, all collected in her bra due to her not bathing or washing it away. She fell silent, took off the rest of her clothes then got in the shower. From start to finish the persuasion took 90 minutes. The shower only took 10 minutes. I was able to help her and spray the water in all the hard to reach places without her being totally naked. When she stepped out I wrapped a big towel around her and gave her a big hug, told her how proud I was of her for trying out something new and frightening to her, and said I hoped next time she would not be so worried about showering.
She has now agreed to take a shower every 2 weeks. Time will tell, but I feel that together we have jumped a massive hurdle. At times I felt like I was psychologically bullying Mum a bit, and I really hated having to do it, but feel it was the right thing on this occasion. Afterwards I dried her hair and made her a coffee then she sat in her cosy dressing gown relaxing, and agreed she felt better. A good result.
The next day when I called round she had got dressed and seemed quite happy. She said she didn't actually enjoy the shower as sometimes she felt a bit chilly (though when I turned up the temperature she complained it was too hot). I guess she would always prefer soaking in a bath to having a shower, but she was doing neither and the bath had to be changed for a shower cubicle because my father can't climb in and out of the bath anymore. I am thinking of getting some extra heating for the bathroom (it only has one small radiator) so that may make it cosier for her next time. Unfortunately Dad has now been admitted to hospital with pneumonia and a urinary tract infection so that is something else to worry about, but just thought I'd share my story with you as it shows that sometimes taking a tough line can pay dividends. Good luck to everybody having similar issues with their elderly parents.