To be annoyed that someone will be sleeping in my bed for a week plus?

Calm down there detective, none of that is relevant to the problem at hand so I'm not sure why you're asking.

What do any of these things matter?
It sounds like you are just trying to criticise how the OP lives day to day.

Friend should definitely stay in a hotel!

What tf has any of that got to do with the price of chips?

Why aren't you saying you're not giving up your bed and mothership needs to fork out for a blow up bed for her guest?

Your bed isn't available* *@BobbyBiscuits.
You have to tell them that, and stand firm. Simply state that you are not able to sleep on the sofa and won't be doing so.
If you want to be petty then camp out in your bed all day and/or put a lock on the door.

"No". Is a complete sentence and that.

They can’t be expected to just come and take your bed! Especially not with osteoporosis!

Is there a room where you can set up some kind of camp bed for her or something?

I think you’re being far too kind here.

This is easy. One of these:

Deal: OlarHike Single Air Bed, Inflatable Mattress with Built-in Electic Pump, Self-inflating Folding Guest Airbed, Inflatable Air Mattresses For Camping or Home Use with Storage, 198x102x46cm(Black) https://amzn.eu/d/03wiPAtA

Inflate it in your mum’s living room (just plug it in. It only takes a few minutes and it’s very easy). Don’t worry about pushing for a date, just say, ‘No worries, just let us know when you have some dates.

She can't sleep in your bed, you need it. Airbed or a hotel.

I just bought one of those, amazed how good it is and it is ideal for this sort of situation.

Have you actually been asked to give up your bed?

Surely your mum understands your physical health needs?

None of which is relevant…

It's not available though is it, because you sleep in it!

Obviously your mum needs to tell the "friend" that she can't stay as there's nowhere to sleep!

Your mum can have whomever she wants to stay in her own home, but that does not mean you have to give up your bed. Tell her that's not happening.

This.
Clear a corner anywhere in your home. Set up a camp bed and a reading/bedside lamp on a small nearby shelf or table.
Welcome your mother's friend for an agreed time, say, one week.

This ⬆️. My children are in a very similar situation, father passed away when they were very young, but it is My House. Hopefully, one day they will benefit from inheriting it but it is not theirs atm and I am free to do what I want with my house. I am also hopeful that they will live independently of me at some point , although, some have special needs so not entirely sure.

No, you don’t give up your bed. If your mum stays up all night then her friend can have her bed.

What’s that got to do with anything

It is not about the bed or arthritis or air bed or whatever - it is about this woman not ever once inviting them to her place in Italy and the mum thinking it is OK to let her inconvenience them. Our family had relatives (!) just like that - relocated to Portugal years ago, before it was even on the Place in the Sun map, in the late 1980s. Where did they stay for their UK visits? Correct. Have we ever been to their place? Not once, not even invited. My parents put up with it and now I put my foot down when my parents are no longer able to drive to the airport, pick them up, change their bed linen, cook for them. I just say (without I am sorry) I am busy with work and my parents are no longer physically fit to entertain you. And I don't even live with my parents anymore.

You’ve got another post that mentions there are 2 others in your house. Who is the third person, and how do their sleeping arrangements fit with this visitor?

OP says they are each other’s carers. Does this mean they both claim carers allowance for each other?

A friend can’t just declare that they’re visiting. If they’re going to be in the area, they can let that be known, but you are simply not in a position to host them overnight - you don’t have a spare bedroom. It’s not unreasonable, nor unusual, to expect a visitor to stay elsewhere if there is insufficient accommodation for them, and any visitor (including your uncle) should be mortified to find that they’d literally kicked you out of your bed. I think your mother is being unreasonable if she expects you to have to sleep on the sofa - adults sleep in beds, and no guest should expect that in order to stay the night, you will have to give up your bedroom.

is my mum for letting her stay indefinitely

This is just ridiculous hyperbole.
It’s your mum’s home, she can have people to stay. She needs to be courteous to you, but you don’t get to decide.

At what point has it ever been suggested that this woman with 2 homes is staying indefinitely?

Your mum herself said it’ll be about a week.
So the guest hasn’t given exact dates - if your mum is fine with that, then it’s OK. Quite often informal visits to friends are fitted round other things - she may just be waiting for plans to fall into place.
As for your mum not being invited to visit… that’s your mum’s choice to accept that. Maybe your mum doesn’t like staying with other people. Maybe the friend had/has a husband who didn’t like it. It’s not for you to decide anything against the friend because of it - your mum is clearly OK with it to have maintained the friendship so long.

All those things are you getting worked up about nothing - I mean, “indefinitely”?
As is the bed situation as you don’t sound like you’ve actually spoken to your mum yet.

”Mum, it’ll be lovely for you seeing X again. I’d love to say I’d give up my bed, but as you know with my conditions there’s no chance. What do you think about this Z bed?”

Hotel, all the way. Why on earth should you be on a couch? She hasn’t hosted your mum for 30 years, surely she doesn’t think she’s staying at yours?