Hi OP.
Firstly you absolutely need to be with your father when he meets the social worker.
They cannot discharge your mother unless she has somewhere to go where her complex needs can be met.
You need to advocate for both your parents and to separate their needs.
Ask your father to raise with the SW that he wants you as his advocate and that no meetings/decisions should be made without your input/support. Be clear you want that on record.
As pp's have said the decision to move your mother back home is very significant. Once there you will really struggle to move her if your father can't cope - and given what you describe I'd be amazed if he could.
On paper it might sound achievable with specialist equipment and carers coming in but they can be hard to source and retain (poor pay). Your mother needs a level of care from your description it would be very hard to provide at home.
You need to ask the SW about D2A ( discharge to assess) where your mother would be eligible for (I think 6 weeks) of funded care. If they can't assess in that time the funding also has to be extended. Has an OT (occupational therapy) assessment been done re: the property and the feasibility of home care?
Ask if any of these assessments have been done and if so ask for the paperwork. Challenge the findings if necessary.
Make sure a CHC checklist (continuing healthcare) has been completed and sent to the ICB (integrated care board). If so was she classed as eligible for a DST (decision support tool) meeting?
If not why not - given your description she should absolutely meet the DST threshold.
The DST meeting is where your mothers SW and a nurse assessor look at 12 domains (eg breathing, mobility, continence, cognition etc) and make a recommendation about funding that the ICB rules on. Funding may be denied, full or partial.
You absolutely need to be present for the CHC checklist if it has not been done and crucially for the DST. Lots of YouTube videos on DST to prepare you and worth a watch.
Re: finances if the property is in joint names then half is her asset. They cannot force your father to sell whilst he is living there to contribute to your mothers care, but if he chooses to sell he may have to hand over her share of the sale (impacting his ability to buy) or sign an agreement with the LA for deferred payments on his death. You need to get advice on this pronto - I'd contact Age UK for advice https://www.ageuk.org.uk/.
Has your mum got an LPA(Legal Power of Attorney - 2 types one care the other finances)? Does your father? If your mother does not it's too late as you can only set up an LPA when you have capacity and it appears she no longer does. Get it sorted for your father now if not done already.
Last point make sure you make clear to your father and the SW that your mother cannot be cared for at home.
They cannot force your father to do this or give timescales - especially if they haven't looked at CHC/D2A.
Hope that helps. 