UK residents can a burglar easily get into a home whose owner they know is out, by phoning a locksmith and saying they live there and are locked out?

Yes, I think that it's possible

I think you have to provide some proof of address?

But more importantly it would be ridiculously stupid because the locksmith would be able to identify the burglar, and have records of where they lived or their bank details.

Some of the dumbest criminals do live in the UK but I don't think they'd go that far lol