Argh, I feel your pain. We were in this position this time last year, except that dd2 wasn't in school nursery yet and we'd missed the application date, yadda yadda.
Ds (then 7, Y3) moved after half term to the new school. I'd wanted to hold on till September (and he was dead against moving schools full stop), but we moved him anyway, and it was a great success. End of the summer term is a lovely time to move schools, as it's nice and relaxed and lots of fun things happen.
If it's the school you want and your little one has a place there for next year, plus it'll save you a 30 min drive, it's a no-brainer. Present it to him as a fait accompli, lovely chance to meet all your new friends, blahdiblah. He'll only miss a few weeks at the old school and the emotion of leaving will be less than if he does it at the end of term. Get him to take in 'goodbye' sweeties for his current class, and tell him he can still visit them. The new class will love a bit of variety at the end of term and will hopefully welcome him with open arms.
Really don't worry. And you could do what a friend of mine did for her ds (I didn't think of it at the time) which was to print a little card on the 'puter with his name and your contact details to give to all the kids in his new class, to encourage the parents to invite him for playdates or to meetup over the summer.
On no account should you risk him not getting a place for September -- honestly, you won't have any probs now that you wouldn't have had in september anyway, plus he won't have the whole summer holidays to wonder what the new school will be like, worry etc. If it does go pear-shaped it's only a few weeks till the holidays anyway. Our position was even worse, actually, as dd2 was still in nursery near the old school so I had a hellish half term with two children in different locations a 30 min drive apart.