I'm in the UK too, and I strongly suspect that he was told just that, since it's the sort of thing that would crop up in an October interview, particularly as his colleagues could have already put in for their leave.
Yeah, it's very possible; I can see that happening too. But the pub's quote that 'All our chefs must work on a Sunday' is a bit daft, because people are entitled to take leave. And if they can't ever take leave on a Sunday - that's a whole different kettle of fish.
Based on my experience of how these things work, there's a rotating shift with some days set in stone and at certain times of the year, it's impossible to get those days off as leave.
I have certain days at work that I cannot miss. In fact, I changed next year's schedule because I have a cast-iron commitment, so the deadline that month has been pushed forward a bit - but I had plenty of notice and checked with everyone possibly involved to ensure that it wouldn't be a problem.
But if i had a job working one specific day a week and I knew I would want one of them off, I'd either talk about it way in advance or mention it in the interview as something that was vital.
His baby did not appear unexpectedly - he was already a father when he interviewed. Why didn't he ask during negotiations? There might have been flexibility in how long he'd have to work on Christmas Day if nothing else.
Also, this pub appears to exist because the community bought it to save it, which implies there's a very strong ethos about it being a community centre. If so many people are eating there on Christmas Day in a very affluent bit of the UK, I wonder what they've planned. Pub landlords in villages don't randomly make the decision to open the kitchens on Christmas Day. They must have a reason.