As you currently live 80 miles away from your mum, I am sadly afraid that her town council won't give a hoot that your landlord has given you notice.
By all means contact them, social services where your mother lives, her GP and Citizen's advice, taking the line with them all that the time is fast approaching, if it is not already here, when you mother will not be able to continue living on her own.
DO NOT say you are willing to be her carer. If they ask you that, say your mother brought it up as the obvious solution from her point of view, but you do not think you can cope.
You have health issues of your own. Do these make lifting your mother, driving a car, etc hard or impossible? Is so, say so clearly.
My own experience is that it will be easier both for you and your mother, if her carer is not her daughter.
Try to get her council to re-home her in a smaller flat and arrange care for her.
Go down to Citizen's advice where you live, and ask whether your landlord can actually evict you at short notice for the reasons given, whether you are entitled to help finding a suitable rented property to move into. Ask them how you will be placed when your mother either dies, or has to go into a nursing home, if you upsticks and move in with her now as her carer.
You also need to know whether you are entitled to a carer's allowance or salary if you take on full-time care of your mother.
You can then consider looking for a new flat for yourself rather nearer to your mother, but NOT within walking distance of her flat. Somewhere where you can reasonably say you will visit after work ( I presume you work) once a week or once a fortnight.
Find out now, when you can retire, and what kind of pension you will be entitled to, so you can plan your future.
Please do not be guilt-tripped into moving in with your mother before you know what it will entail, both now and later.
I am 14 years older than you, and have seen far too many of my own generation assume the unmarried (or married) daughter's burden of care for aging parents, getting no thanks for it, either from parent or society, and left worn out with no friends - they disappear fast when you are looking after a cantakerous old woman or an Alzheimer's patient on your own.
A paid, unrelated carer can either say she won't go back to Mrs N. (as one of my mother's did) or as another did tell her she was being difficult for the fun of it, and to snap out of it. Any daughter who does the like suffers agonies of guilt.
Don't put yourself in that position!