I have just visited a lot of these schools this year as I have 11 yr old twin with ASD.
Blossom House was already full last year for this year's September intake when we applied and we are on a waiting list of 14. They don't really like ASD children is my impression but have good speech and language provision.
Eagle House have a new school opening nr Sutton. It will have 2 sections - one for less able and one for very able. They weren't very helpful and said my boys wouldn't fit into either category really and that wa only from looking at the paperwork and without even assesing the boys so I didn't think that was great.
The Priory School in Roehampton looks very smart but is very, very expensive. The head teacher is very experienced, used to be at Heathermount in Surrey. Think they are aiming for quite academic children.
There is also St Dominics in Surrey - academic but ASD specific.
There are 2 More Houses - one is boarding in Godalming and the other is in Farnham and totally unrelated. the first is speech and language (we didn't look as it was boarding) and the other takes quite able children - my boys didn't do well on their assessment day but our middle son subsequently did a day's assessment and has been offered a place (he is 8 and higher functioning than my twins).
My twins were offered a place at Centre Academy and will be starting there in September.
Good luck, its so hard finding the right fit for each child as the schools definitely take varying levels of language/behaviour etc. Best to look at them all if you can.