Sounds like a nightmare and I would be none to happy to have the car taken off me simply because the number one tyre changing company in the UK has not got a new tyre to fit.
It’s not something a disabled driver can foresee and my experience of going to KF when the tyres look low on tread is to get the knock back and told to come back when I’ve got another 500 miles on the clock.
I assume you have one of those stupid cars that come with no spare.