Thinking of you all this morning and hope he is as ok as he can be. Does he have Attendence Allowance? I think Rehabilitation goes under the term Reablement here and I think the mobile Physio your Dad has might come under that team. They should Aldo have had an OT visit to ensure they have everything he needs e.g. Grab rails.
It might be an idea if they don't already to get SS involved, tracking down the hospital SW would be a good start and they would then get transferred to the adult team once home. Your Mum is entitled to a carer's assessment as his career to check her own needs are met. Not that I think it will be particularly worth doing but you never know.
I think you will have to be very firm about making sure an adequate care package is in place. They tried to duscharge my friend's relative from Bournemouth who has severe dementia and whose partner had started passing out due to the stress. They blatantly needed a CH placement so their daughter refused and realised they were eligible for CHC so awaiting for that and a CH placement a month later.
Unfortunately resources are hugely overstretched here as we have a larger elderly population as people retire here but no allowances made in funding terms for that. Something one of the surgeries locally has started is having a paramedic attached to the surgery who will go out. Ours had what they called a tracker nurse whose job it was to keep those vulnerable to hospital admission from being admitted, she was amazing with my Mum.
We used a private Physio for my Mum when she was bedbound ad had to be hoisted. She went from that to being able to make a long haul flight to Asia to see her new Grandson. It was expensive but fortunately she was able to afford it.
I'd also recommend of keeping a diary of things that happen ,it can be very useful to look back on it as at the time you're knackered dealing with the emergency at hand.