*Will I be made Homeless if mum dies*

do not move out of your present flat.
wait until they get a court order for eviction.
that will mean you are officially homeless, and in a better position to negotiate with the council, whether in your area or your mum's, with a view to moving with her to a suitable 2 bed flat.
she cannot manage stairs, so would need to move probably at some point. stair lifts are not always the right answer. some people cannot safely use them. also not suitable to get to a loo in a hurry.
contact shelter for specialist advice.https://england.shelter.org.uk/housing_advice

Like some have suggested ... get in contact with your moms Council, as I don't think you would be entitled to live there also someone else suggested about getting a flat , or bungalow, even a warden controlled flat or bungalow , as she will still have her independence, has she got an alarm that if she needed help in an emergency that would help aswell

Definitely contact the council from whom your mum currently rents. They are the only ones who can give you this information. You need to be very sure of your position before you make a decision.

The question is - do you want to be her career?
Answer it without guilt or sense of responsibility

As your mum has trouble with the stairs, could you ask the council if they have any single bedroom bungalows, so bedroom, living room, kitchen, bathroom, and ask if you can have joint tenancy, explain the circumstances, then your mum has the bedroom, and you use the living room as both your bedroom and as the living room, then if and when you are the only occupant, you would be one person in a single bedroom bungalow. These single bedroom bungalows are often occupied by elderly married couples, so two people living in the building would not be overcrowding.

I don't know if that is possible, just wondering.

It might help if your mum's doctor or a nurse from the doctor's practice supported an application to do that.

In fairness, as your current landlord wants you out at some as yet unspecified time, you don't need to stay there until they say so: you are going to have to move someewhere anyway.

I hope this helps.

As your mum is in a 3 bed that she doesn't need, would she consider downsizing to a 2 bed flat/house so that you could claim together for?
That way you would be a tenant!
Look into this as the councils often pay to help you move etc when downsizing as they need the places with more bedrooms for families with children!!

I’d seek some more information from Citizens Advice before you do anything. As your mum is occupying a three bedroom house my guess is that is highly unlikely that you’d be allowed to succeed to the tenancy as the house will be needed for a family. However the council may have an obligation to find you alternative accommodation for a single person at that point? There’s a lot to consider though, as presumably you will have to leave your job and your friends too, you have to think about your long term future as well as your mum. The council may well consider alternative, more suitable accommodation for her if she requests it?

I can’t help you with any advice, but I’m sure there will be someone who can. All the best and look after yourself.

I suspect that carer or not, you would have to leave if your mum died, but perhaps you could contact the housing department of your mum's council to ask about this.

My previous neighbour, in a housing association property, remarried a year after his first wife died. His new wife could not have stayed in the house if he died. However, as he had health issues, they were able to get a transfer to a different, more suitable property, and the new tenancy at this property was then in joint names.

Get in touch with your mums council and social worker and see what they have to say.