In was wondering if there's any chance of your mum being in a wheelchair for the day centre. I'm sure places like the Red Cross would rent one out so you could give it a trial. Could you have her assessed by an occupational therapist?
if she spends the day there and has lunch there and there is no-one at home to fill the gap I think they are in a dodgy safeguarding position doing it without any notice. I think you will find that most organisations hold the veto on where they draw lines on whether or not they are equipped to cope with people.
It also seems very unkind to do it right now just before the Christmas festivities are about to kick off - it would be kinder but no less difficult if they said 'after the Christmas break, in the new year' and it would also give you time to find somewhere more suitable.
We had this problem at one stage but I am afraid I fought them tooth and nail because I felt that they were being unreasonable and the backtracked and my Ma was able to stay right the way through. I'm afraid i wouldn't accept it with no notice at this time of year - they could be leaving some vulnerable people in a very tricky situations so I would fight it on that basis alone