afaik, when you move county with a statemented child the statement moves with them, and holds until a new assessment is carried out.
again, afaik, a school can make an exception on numbers for a statemented child (assuming eg 1:1), as the 1:1 take sthem out of 'class numbers'
so, technically there should be no reason why your ds isnot at school.
I would be phoning the LA tomorrow, telling them the school you prefer, and reminding tham it is their legal obligaiton to provide a suitable school place for your ds. and keep phoning to chase. eg if you get fobbed off with "can't do that unitl this afternoon", then call back in the afternoon.
call daily, be polite but firm, and keep reminding them of their legal duty. do not be fobbed off. write to them following each phone call (email is ok) to 'clarify' what has been said - keep the paper trial going. so, if you are told 'it could be some time' write clarifying that you have been told there is no suitable school placement at the moment, and asking when there will be a place, as your son is legally entitled to an education in school.
if they persist in saying that there is no place, or it is difficult to sort out, start researching independent schools (whether you want one or not). find one with a place, that is suitable, and ask for it to be considered, since the LA hasno places available.
this shoud ensure it gets sorted out asap.