In a pickle - mum in hospital, won't be safe at home

Speak to the hospital social worker and your local adult social services. Also ask about continuing care ( you can ask for an assessment). If they have savings over £23k they may need to self fund

Please make the hospital aware that you are extremely concerned about her safety if they try to discharge her back to her home, she will certainly fall if she's trying to walk. And that she is also having cognitive issues.
Also that your father is frail and unable to care for her himself.

The hospital should enquirer as to her living arrangements and will not discharge her to go home as long as they are aware of her true situation.
She should at least have a 'care package' with carers coming in each day to get her up, washed, dressed etc.
The hospital will be especially cautious after a leg amputation . She may be sent to a rehabilitation unit in the first instance, it sounds as if it will be necessary.
It sounds to me as if she may be best cared for in a nursing home.

Problem comes as she has capacity she can discharge herself and tell them she doesn't need any help... sadly.

I don’t know enough to advise you OP sorry but when my Gran needed support Age UK were brilliant. You’d be able to Skype their helpline next week and they really know their stuff.
If I were you I’d also make sure the hospital are very aware of your Dad’s inability to care for your Mum and that you are overseas. If they think there are people that can offer support even if that’s a neighbour they will often try to discharge. They are desperate for beds and there is a shortage of carers at the moment right across the UK.
It sounds like your Mum will at the least require respite and rehab in a care home or even maybe a nursing home. With the right support in place and caters visiting regularly she may be able to then return home, but often the elderly end up permanently in care/nursing homes after what was planned to be a short stay following on from a hospital admission.
The hospital will have an OT that will be involved in deciding what needs to happen on discharge so I’d ring the ward and ask who you need to speak to about plans for discharge, plus speak to Age UK and take it from there.

As she is in hospital, she should not be discharged without a needs assessment. If it is decided that she can cope at home, social services should provide a six week package with up to four support visits per day for free. Thereafter she would be expected to pay depending on assets - means tested.

It may be that the amutation means she would need to go to a specialised rehab facility first but the main thing is that she should get a support package.