We were in this situation last year. The house purchase dragged on for so long (sellers were stalling because of an onward purchase they hadn't told us about - claimed no onward chain - as PP said, anything can happen) that we missed the application deadline, plus the late deadline, for the new school.
When we exchanged, I applied immediately (within 20 mins!). DC1 was offered a place as the school had several places available in her year, and in the years below - but DC2 was not, as unexpectedly the school was oversubscribed for YR and all places taken. We were allocated a school several miles away with a bus ticket provided.
we appealed, and won the appeal, but it was very stressful. Without the sibling already at the school, we'd have had virtually no chance.
Clearly we were both unlucky and lucky, but Tiggytape's 'freak years' do happen. I'd advise that it's best to exchange before the application deadline; if that can't happen, either rent locally and apply from the rented address, or postpone exchange until AFTER the offers date for primary places. At least then you'd know if there are places at the school you're about to spend hundreds of thousands of pounds to move near. Or have a private school as backup. I'd also second looking at the schools. I toured many - and we ruled several villages out as a result, despite the schools being OFSTED Outstanding. They just wouldn't have been right for my DC. The school we were allocated was one that we'd dismissed, which was particularly galling, but at least in the appeal I was able to argue exactly why it was wrong for my child, why they would suffer psychologically and why the one we wanted would suit them so much better.