For state special/specialist schools, you would need an EHCP.
Without an EHCP, you will be limited to whichever mainstream schools have places. For that, you would be best starting by asking you LA what schools have places. (Or appealing for one who doesn’t have places and refuses a place in the hope an appeal is successful.) Although, personally, I wouldn’t move whilst going through the EHCP process. If DD can’t attend school, alternative provision should be put in place. If that isn’t in place already, IPSEA has a model letter you can use.
If you get an EHCP, depending on where you are, you could look at Orange Tree, Egerton-Rothesay, Odyssey House, Canbury, The Rise. They will probably be too far out, but if not you could look at The King’s school, St Christopher and Red Balloon. Some are independent MS, some are technically mainstream but not conventional MS, and some are SS. Some will be too far, but it depends where you are.
If the LA refuses to assess or refuses to issue, appeal. School support is helpful but not essential.