I asked at Sancton Wood and they were full. Also it's worth reading the mumsnet local threads about them as I understand they are quite overwhelmed there and things are a bit chaotic.
re: doom and gloom, I think the difference between the north and Cambridge is house prices. Cambridge has london house prices but not London weighting on teacher's salaries which makes living here extremely challenging for teachers.
I gather that you really have to be solidly religious to get into St Bede's but I understand that a child with an EHCP could get a place there if it was named on the EHCP.
The other thing to be careful of in this country is that children are "entitled" to certain things, but the government doesn't provide adquate funding for the schools to provide those things. So teachers have to say that x, y, and z will be provided and could get in trouble for saying otherwise. But in reality the staffing is just not there. So if you are looking for a very specific situation for a child, it might be hard to get straight answers about what is actually feasible.
There are lots of articles in the press at the moment about a massive uptick in the number of EHCP applications and how the councils are actually declaring bankruptcy because of it, so it is a serious problem.
Sorry, I think I'm providing a lot of the doom and gloom on the thread, but I'm right in the middle of it at the moment and finding literally no good options and only very few people who will actually level with me and say what is actually feasible.
I think realistically the way forward for the OP is to be willing to pay £££, and talk to alternative providers or some of the new providers who are setting up to plug the gaps in provision.