I have similar issues with my elderly parents so can sympathise with what you're going through. My father is 82 and has never been very keen on taking baths - even when he worked full time he only took 1 bath a week (although he did have a thorough wash morning and night too). My mother is 84 and used to love soaking in the bath a few times a week, but now it's a battle to get her to take a bath at all. I reckon that the pair of them have probably had about 3 baths each over the course of the last year - which is about 3 more than either of them thought they needed!
My parents don't seem to be washing very often nowadays either, but when they get a bit smelly they will at least agree to change their clothes and sometimes I can get my mother to wash her hair at the sink or soak her feet, with help from me. My mother is frail but has no real mobility issues - she's just given up on life. My father has dementia so loses track of when he last bathed/washed/changed his clothes. Also he has hip/knee problems, so he struggles to get in and out of the bath (but won't let anybody help him). The other day he was really quite smelly and wanted me to take him out to the garden centre, so I used that as a lever to make him have a wash and change his clothes. That was when I noticed, as he stood over the sink having the briefest of washes, how breathless and tired he got, just standing there for a few minutes. That was when I had a long conversation with both my parents and said I think the time has come to get a walk in shower cubicle with special seat, so they can both have showers and not struggle in and out of the bath. They finally agreed (they were resistant to this idea in the past, saying they don't like showers). The shower cubicle will be fitted in about 10 days' time so it will be interesting to see if they make use of it.
Anyway, back to your mother and the ongoing problem. When comparing her situation with my parents, the one thing that jumps off the screen at me is the fact that she is regularly walking around the village etc and still socialising with people. It seems she has no real mobility issues and so the lack of washing/bathing seems to be more a case of won't rather than can't - in other words, it seems to be something of a lifestyle choice for her. Not very nice at all for anybody who has to be around her for long, and not at all funny for you and your wife, esp. when you are both making so much effort to help her. OK, so she's made her choice - now you need to make yours. Choose NOT to have her in your home or your car, unless/until she changes her ways and shows you a little more consideration. Don't feel guilty about it - your mother is not disabled and has energy enough to walk around and take bus journeys, but seemingly just chooses not to keep herself clean. Be firm with her on this issue. Let her know you will continue to offer her help in various ways, but explain that you are not willing to have her share your personal space - home, car etc unless she makes an effort on the hygiene front. Explain how strong the smell of her BO is - old people often lose their sense of smell so she might be quite surprised to hear this from you. Your mother has every right to choose to be a "dirty mare" as she so quaintly puts it, but you have every right to restrict contact with her and restrict the impact of her lifestyle choice on you.
PS: Just realised I'd missed reading your post on page 2. Noticed you said that 2 or 3 times a week she turns up in the evenings and just lets herself in without even knocking the door.
It sounds to me as though you've made yourself too available to her and she's taking advantage. You really need to change that. I suggest you have a long conversation with her where you explain all the reasons why the open door policy is going to end. Tell her that from now on your doors will be locked at all times because you are entitled to your privacy and she is not showing any consideration or respect for it. Just because she's your mother that doesn't give her the right to have free rein of your home, particularly if she chooses to be a "dirty mare". Your home should be your sanctuary.