It's mine, I tell you. My own. My precious.
Do I have the keys? No. Can we pick the lock? Also no.

I contacted a company who stocks spares for UK payphones, asking them to send me a set of keys to try.

They replied with "That will not be possible" and the following photo.

@jonty I can open it for you without damaging the box too much and then you can replace the lock cheaply.
@jonty oh holy crap I literally didn't even see your phone box til after I wrote my previous response... I can help even more. If you want. I can probably visually identify the manufacturer of the lock and the aftermarket keyblank that would fit it. You could pass that along to the locksmith.
@jonty Tell them it's a small format single milled key, not a tubular key. They will say "OHHHHHHH.... why didn't you say so?!" They're showing you an image of a wall of keys for commercial payphones that were installed by telecom operators. The phone box you posted looks like a model someone would install themselves in a automat or something. OR it was later converted for that use case. Probably.

@Netraven This is the payphone that was installed in public phone boxes across the UK by British Telecom, the nationalised telephone operator.

The wall of keys was for different models than I have here - because I am not supposed to have this phone and they assumed I had made a mistake with the model number...

I've got plenty of locksmith help on the ground, but thanks for the offer!

@jonty @Netraven
That exact model was also installed in public phone boxes across Ireland by our national monopoly PTT, shortly before public payphones were opened to competitors.
Every operator had their own distinctive payphone and kiosk design, it was a wild time.