Mom gets hair done once a week. She can’t reach the back of her hair to comb it, and after 3 days it’s awful back there. Advice?

Your profile is confusing. Looks like Mom was in a NH but finances brought her to your house? Your husband is with his mother and his sister looks after them.

If Mom was living in a NH and ran out of money, why did you not apply for Medicaid. But then your question is really about Moms hair. A towel will not help IMO. Years ago woman used satin pillow cases to keep their hair from getting messed up. If you can't raise ur arms, sit her lower in front of u so you can get the back of her hair.

I am going to ignore the hair part of your question for a moment.
You are caring for your husband at home with dementia.
Your mom is in a skilled nursing facility at this time and you are thinking of moving her into your home.
I hate to break it to you but Superwoman does not exist.
If you have to apply for Medicaid for mom to keep her in the Skilled Nursing facility do so. You can not care for 2 people with dementia. You would have to hire caregivers to help. (Hopefully you have caregivers now to help you) but 1 paid caregiver might not be enough to care for 2 people with different stages of dementia at the same time. And it will not get easier, you mention you have mobility issues (problems lifting your arms).
Now for the hair....
Some of the "satin" pillow cases are supposed to keep hair less messed up.
I would go for a cute short cut, think Jamie Lee Curtis type pixie cut.

If Mom has limited shoulder movement for hair brushing, wondering how she bathes, dresses & all the other tasks?

Is she living alone, with you or in AL?

I'm with Polar & 2nd a short hair-do.

Hmm... The only thing I can think of is to have her hair cut short. My mom has her hair cut monthly, pretty short, but still stylish and feminine. She doesn't need to comb her hair daily and it still looks fine. A bonus is that her hair is easy to wash and dry, too.