I too wash with cold water and I hate it but it's the best/fastest/cheapest or whatever you call it way. I rather sit in a hot tub though but the house is too cold and at the time that tub is full the water is cold.
I use hot water bags too, we all do. We take one in bed if needed and at daytimes if it's cold one on the lap and if you place your hands on it you can warm them as well. It also helps to relief pain so it's a good and simple thing.
If my feet are very cold it's faster to put them in a bucket with hot water and that's perphaps what you should do too if your wash with cold water.
If the sun is still warm I wonder if a tank or black bag filled with water and warmed by the sun can help showering. Can be it's not enough but at least the water won't be that cold. They use those bags also for camping, no need for electricity and such.

I use something like this they last for many years and many people use it. Just filling it with water can be a bit dangerous if there's air in it.

camping shower might be a solution and there are different types.

I don't think 30 minutes in the bathroom is long. Those I know all to seem to be there for ages unlike me. That endless shivering in bed is terrible as well.
Although you say that you can handle the cold and still manage I do believe you should see if you can find a solution to have a bit warmer water because at the time you can no longer stand it, it might be too late and it does affect your health. I also think it might help your mom by not always putting the kettle on and taking care of those extra preparations.

I wonder about the type of plastic bag you use to fill with hot water since the plastic we have here will not keep the water inside and I wouldn't know how to tight it that the water won't escape.
By the way for the animals in the shelter I ran I used bottles (glass) and filled them with hot water and wrapped a cloth or pillow case or diaper around it like indeed is done with babies. It did help to keep them warm or at least warmer than without.

I wish you a warm day/night without cold water.