st johns is hardly really a church school tbh, it's just the junior part of hampton wick infants
sacred heart has a link to teddington, I would guess based on location (as both it and collis are the second closest schools to teddington after st johns)
if you look at the booklet they sent out this year (dd is in yr 6!), no-one got into Teddington who wasn't at a link school and even some children in link schools didn't get in due to their distance from the school.
If you think that the primary schools in that area have all increased - collis now takes 4, stanley is taking 3 (think it might be going to 4?), st johns 2 and st mary st peter I think is about to take 3, and yet none of the secondary schools have got any bigger. There used to be a big outflow to private at 11 but that seems to be slowing down.
Don't think st james is close enough to any of those schools? I saw one of my old playgroup friends whose child is now in yr6 at St James and she is beginning to fret.
Just fyi, the private of choice over the last few years seems to be Kingston Grammar!
If you have a girl at st james, I know quite a few catholic families try for Gumley in Isleworth.