Oops.
Legally, your child has to be in full-time education in January. If you have robust childcare in place - e.g. a childminder or a nursery used to children starting school at different points in the year & not all leaving in July / August - then you can keep him there until then.
There may be a little movement in waiting lists after September (as some children don't turn up to take their places e.g. through moving or going private) and you can sometimes be lucky as someone may be on the waiting list ahead of you but decide not to move schools once their child has started elsewhere. However, in some areas, waiting lists are 'wiped clean' in September as it is assumed that everyone is now happy in their allocated school, so if that is the case where you live, you need to make certain that you remain on the waiting list after September.
You need to get on the 'phone to the LA on Tuesday, and find out whether there are any schools with vacancies for September (these are likely to be poor schools, but may e.g. be more convenient for you to pass on your journey to work), and also to place your child's name on EVERY SINGLE waiting list that you consider remotely acceptable (as the place that you have turned down may well have already been given to another child).
And you need to start planning now for what you will do if, come January 1st (or September, if your childcare cannot manage the extra term) your child has no school place. As others have said, you will have caused the situation and so you will be deemed to be to blame for your child not meeting the legal obligation to receive a full time education - the defence 'well, I didn't like the place I was offered' is not acceptable to courts.