OP look. Some posters on this thread don’t actually know what Alzheimer’s is like. The ones that do have made helpful posts but they are the ones warning you not to let your father proceed with an operation.
Bottom line is if your mother is at the stage that she is not recognising relatives and thinking she is going to prison she is in no fit state to have any kind of operation. Full stop.
The poster who knows someone who had “elective” surgery with Alzheimer’s appears not to understand it’s a spectrum and in the early stages only features mild cognitive impairment. People with Alzheimer’s can still have mental capacity to make decisions.
Based on your mother’s symptoms how likely is she to pass a test for capacity to make medical decisions?
Saying your mother is on the same page as your dad is meaningless, my father, for whom I’m a carer, simply agrees with whatever people tell him as he has no real idea what they are saying.
The incontinence is most likely caused by the dementia, in which case an operation will not make any difference. It will not in any way magically make your father able to care for your mother. I mentioned my aunt’s incontinence was not caused by dementia, but even so she was still advised not to have any invasive treatment.
You MUST look at the terms of the LPA for health and welfare. Does your father even have LPA? If not he has no legal right to make any decisions for her and if you let him and it goes wrong you will be at fault. If he does.- is it “jointly” or “severally”? If jointly then your father cannot make a decision without you and the other attorneys’ joint agreement. If “severally” your father can make a decision on his own but he would be very unwise to act against the wishes of any other attorneys.
As you have LPA I would ring your mother’s GP and tell them you are extremely concerned about your mother’s condition, her capacity, and your father’s capacity to care for and make decisions for her. (Even if you didn’t you could still discuss your concerns about her care).
The private hospital aspect is a concern because, as can be seen from cases of inappropriate plastic surgery, some private surgeons are happy to perform surgery as long as you pay them.
Even if this is an NHS decision and operation, that doesn’t mean it’s in your mum’s best interests. GPs are not consultant geriatrician or neurologists and if your mother’s Alzheimer’s was diagnosed some time ago, the GP may not be up to speed with exactly what stage her dementia is at.
You need to wise up and start using your LPA for your mother’s benefit. Sometimes care from a relative can be, however loving and well meaning, inappropriate. Your DF is totally out of his depth.
What your mum really needs at this point is not an operation but night-time carer.