Is the only way to give up my sitting room?

I think your best bet is that she presents herself as homeless to the council. She will be put in a hostel before temp housing. They won’t see her as homeless with her living on your sofa.

Could she find something part time that can be done from home. There are some jobs that might be able to give flexible working of that nature. Otherwise, if she genuinely can't work I presume she would need to be assessed for disability, which might make her eligible for support eventually. Could you evict her after the 3 months is up on the basis you are overcrowded. Would that get her a foot in the door for emergency housing? Not the best way to go about it but you know you can't keep her with you permanently.

What about an air bnb rental for a short term, does she have enough left of savings to tide her over for three months and also potentially look for a work from home job at the same time?

OP could you say what area you are in? People might have more specific thoughts?

She is unlikely to pass the habitual residence test immediately upon her return from abroad, she will have to reestablish herself in the UK to qualify. I assume she holds a UK passport? Upon her return she needs to claim UC, if she is awarded that turn she might then be able to argue that she is habitually resident. Just for info, it's not actually how long the applicant is back. The test is ' have they reestablished themselves in the UK'. So clearly having a job, bank account, doing all of the things that a resident do as opposed to a visitor.
She would then have to be homeless and in priority need to be offered homeless accommodation by the local authority.
I realise it's a difficult situation and this might not help you but it does explain some of the difficulties.

I think that you need to bring her to your house, then weigh up options. You can't secure any type of housing without her being in the country and surely you can't see her street sleeping? You could contact a homeless charity and get advice. The women's hostels can be as violent etc as the men's.

I think student houses are the one exception now where you can rent a room just with a guarantor and no job, don’t know if she could just pretend to be a student? Or even enrol on some kind of short course in order to get a student ID …..

Could she be a lodger somewhere? Thats usually cheaper than most forms of renting, and as you live with the landlord if she meets somebody she can get on with then they may be more sympathetic/amenable to your mum guaranteeing the rent while your sister waits out the “habitually resident” period.

A lodger that would mostly be at home so could also do dog or cat sitting for example may be appealing to some people? Depends on your sister’s health obviously.

If any friends or family have a spare room they might be willing to rent it out on a lodging basis as they would know it’s short term, and lodgers can be evicted easily if they want to.

It is enough, it's all that is needed for shared houses for students so there won't be higher required for anyone else. You also don't have to go through an agent, plenty of private house shares as people need the extra income for their spare room

Edited as update from OP makes it sound like she isn't a citizen which makes my comment inapplicable.

My disabled brother was recently in council emergency accommodation, then placed in supported living/shared housing run by a local housing trust. The emergency accommodation the rule was no more than 24 hours/1 night away from the accommodation at a time - obviously they don't want to be paying for accommodation for people that have somewhere else to live. In the supported living they're much more relaxed and he's been to stay with us for up to a week at a time which is fine so long as he lets the staff know. So OP likely could have her sister stay on the sofa for the odd night at least.

OP I think you can't offer to house your sister permanently. I think if she really can't get a private rental or room in a shared house you need to let her go into emergency accommodation. I think on that front the person you were speaking to may have been a bit alarmist, there isn't a vast surplus of such housing but it is available, my brother waiting about 12 weeks then was placed (probably could have been placed sooner if he wasn't fussy about location. If your sister is eligible for over-55s housing or supported living/sheltered housing I'd be surprised if the council can't find her somewhere. Depending on where you are in the country a room in an HMO or a small private studio may be quite possible too. Based on my brother's experience the emergency accommodation is fine, not ideal for a disabled person, the room itself was perfectly comfortable (just a basic hotel room) but there were very very limited cooking and laundry facilities - hence him frequently coming to our house to wash his clothes and get a decent meal...

Have you looked at gumtree/free ads section on house shares. We have a lot in our area and not all with an agent, if you and your Mum could cover the rent until she has sorted out her money situation etc that may be a solution?

Would she be allowed to stay on a pull out sofa at your mums- even if just weekends? (not sure of sheltered accommodation rules?)
I too would be worried she may not be entitled to claim support/NHS/benefits etc.
Shelter might be able to give advice.
www.shelter.org.uk/

Not enough cost wise do you mean? Can your sister work at all? Can she claim housing benefit? She could always go into a property owned by guardians they still pay rent but I think it’s less than in a house share.