Gus - is your most convenient school undersubscribed at the moment? If so, it's worth putting it down as your first preference. If it's oversubscribed you probably won't have a chance.
When my first dd (who is going into secondary school next year) was starting primary school, she had got a place at our nearest school, which I was over the moon at, then in March we were told that the landlord was selling our house and we had to move.
I found one a few miles away. She could still have gone to the school we had a place at, but there was another literally 2 minutes' walk away. This was in May. So, on the off-chance, I rang the near school and explained the situation, and asked if they had any vacancies.
Amazingly, they did, and I was able to simply switch her to the new school, and she has been very happy here.
You could try that - sort yourself out a place FIRST - then ring your preferred school and ask if they have any spaces. You might be surprised.
This time round I have one dd starting secondary and one dd starting primary in the same year!