Hi,
Sorry to hear of your wife's problems and stress. My circumstances were kind of similar at the time Slough's finest chose to pick an easy target. Ironically I was parked in a parking bay to attend a physio for stress related neck pain followed work + my mother's passing.
Anyway, I am not qualified to give legal advice but everything you have said here sounds wrong. The council parking attendant has to physically put the yellow notice ON your car for it to be "served" and thus legal.
If they have no proof then they don't have a leg to stand on. Yes, the ticket could have blown off in the wind but that's their fault. They have a receipt system to avoid arguments but it seems they are can't provide proof so can't argue.
Definitely do NOT give the car keys to anyone or even any money. That's absurd. I would seek professional advice ASAP. Try *all* or as many as the following:
(a) citizens advice bureau
(b) a soliticitor (15-mins free basis)
(c) online legal help, such as
Legalservices Shop
(d) the Samaritans
(e) the council (again) explaining you feel threatened
(f) your local MP
(g) the police about the ticket that doesn't exist AND the abusive bailiffs
I know the last one sounds a bit daft but I had a friend in desparate trouble recently and they were excellent and gave us several very useful phone numbers. They even ring you back if they don't know immediately.
The debt collectors sound very aggressive and sound like they are intimidating you. They are only allowed to recover money where owed. Your circumstances are debatable so it is for a court to decide IF you owe anything not some bully.
I know you will be feeling angry/upset/anxious and many more. Take a DEEP breath. Dig deep and get on the phone and keep talking until someone listens.
I hope that is of some use. You don't deserve this, even if your wife DID over-stay. Somehow I doubt she did.
Good luck.
"Victor"