Family member moved in….

It’s not harder for them if they are applying with a social landlord that does housing for 55+ If they are competing against professionals for rooms then yes.

Give them a deadline to be out and when the day comes if they don’t have a place, then still stand firmly and tell turn they need to go.

Has he looked into being a lodger rather than looking for a room share in a house. It might be more suitable since he is older.

Are they over 60? If so, have a look at Alms Houses.

Close to zero chance of 55 year old bloke getting a space in a shared house in London.

My son and his gf (early 20s) ended up taking a shocking room in an absolute dump without having seen it because they'd missed out on so many and time was tight.

Commonly it is existing tenants who have a say on housemates for house shares (even if the rents are separate, landlords often leave it to the tenants to do the viewings and in return the tenants get a say in who is picked) I can't imagine there are many who would be looking for a 55yr old male housemate.

Also if lettings are anything like Manchester, good places can go extremely quickly. Whole flats are let without people even viewing, house shares tend to go v quick in the city if the price is decent. Some sites hide the ads behind a paywall for the first week, so you need to sign up and pay in order to get the contact details - by the time the ad becomes free to view, it'll have gone.

None of this is your problem to solve though! He's had plenty of time to sort himself out

A single male has zero chance of getting a council property in London

They could even lodge somewhere else, and could look outside of London too. I suspect they’ve just got too comfortable with you and will keep on failing to find other options as long as you let them.

Why on earth would you put up with this beyond the initial short-term period? If they’re not SEN or got some kind of reason they need your support in that way, why can’t they manage their own life and housing like every other adult has to? They’ve saved a fortune by living with you for five years. Why can’t they rent their own flat, a studio or one bed or small house rather than a spare room? Their age may well be off putting for house sharers but as you say this person qualifies for over 55 accommodation so it shouldn’t be some insurmountable task to sort that. They shouldn’t need you or the council to come to their aid. Be clear and direct - you’ve done more than enough. Make a deadline and stick to it.

It’s reasonably easy to find a room in a shared house in London if you are young (so student or just starting out).

most shared houses fill a room either through friend of friends (Helen’s boyfriend knows someone) or they interview.

older people (and in this context over 35 is older) have much more trouble because they don’t gel socially.

they’ll need to look specifically for older shared houses or a lodging type situation.

the “interviews” can be difficult to pass.

OP you rescued them and are continuing
to do so

you don’t have to help them find somewhere to live, that is their job, and you don’t have to give them 6 months notice either, they’ve had 5 years!

tell them to go in the next month

please do not continue to rescue them by accompanying them to see places

OP, you've been far too nice for far too long.
Your lodger has become complacent and entitled, and will probably try emotional blackmail to stay.
If he/she were ever serious about moving out they would have been actively looking and saving for a deposit during the past five years.
You need to get tough and reinforce the six month notice with constant nudges.
Yes, it's hard to find affordable accommodation in London, but if they can't afford London they need to look further afield.
Don't fall for the sob stories and excuses - your ''short term'' lodger has had five years to save and sort something out and hasn't.

From what I understand, you need to make them homeless to get a council place, or it’s a long wait.

Also, unless you give them a deadline, there’s no incentive for them to move out. Tell them they have to be out by July 1st (or whenever).

You also may have to get tough and sit with them looking at spare rooms, and go with viewings with them. They don’t sound very proactive themselves..

Don’t be a guarantor for them. You’ll end being responsible for the rent when they don’t pay.

Your first sentence is why people are wary:

Family member moved in with us 5 years ago as he fell out with previous landlord, also family member

They are absolutely playing you. You could arrange and accompany them on these visits to see for yourself but honestly unless you are tougher and literally make them homeless, the will never move on.

Who are they? In law/sibling? Why is it on you to provide accommodation?