Jesus, sorry to hear that for your mother-in-law @Bluewolf. What an absolute palaver on all aspects. Shame the Fire Brigade didn't know about the key, surprised they didn't just bring it down with a big red key.
I'm amazed Kingston Council say it takes over a month to get a lock sorted! I'm guessing she's in rented accommodation by the sounds of it.
I'd persist with speaking to them and constantly letting on to them it's a safeguarding issue for a frail, physically vulnerable lady who clearly has care needs before the fall has even happened. That's obscene to think it takes best part of 28 working days to get a lock change. I work in housing albeit not for a local authority, but there's bound to be a department that can arrange for emergency lock changes within 24 hrs. Shocked if there isn't. And given its a communal block, it's their responsibility to ensure the safety of the block and get a fix in within a certain time of it being reported.
Alternatively, I'd get the flat door lock fixed yourself to a better standard with a better function, and then stick a complaint in and ask for you to be reimbursed. You're well within you're rights. While you're at it, let them know you'll be taking it to the Housing Ombudsman if they don't reply or repair in a more reasonable time.