Hi moony, I was in a similar position to you last year and was advised by the Council that I was better off arriving after January and making a late application, than arriving a year later. This was confirmed by staff at Collis and Hampton Wick, who both said that each year they have a large number of children who give up their places in early September in order to take up private school places. In the end, we applied late March and got a place at Collis by the summer. BUT this was in the bulge year: I've met kids who moved to Teddington in August and still got a Collis place. I don't suppose it will be that easy for you, but Hampton Wick didn't have a bulge and they still seemed confident I'd get a place.
Good news is that there are five great primary schools in Teddington. Collis seems to have lots of snob value (and is indeed brilliant) but the others are all good too. Hampton Wick, for example, is a lovely school: small with a real family atmosphere, a bit more socially mixed than Collis, but still producing great results.
It's a bit early for you to think about this, maybe, but if you move near Collis or Hampton Wick you will be just a few minutes walk from Teddington school, a mixed secondary that is a pretty good school (decent school + walking distance = London gold dust). But if you live further north, nearer Stanley Road, you may get in the catchment for Waldegrave School, a really good girls state secondary.
DO move to Teddington! It's lovely here. Not for you if you like urban grit: it's almost like the Truman Show, it's so safe and warm feeling. I've never lived anywhere so friendly and comfortable, and the place is DROWNING in mothers and young children. I've lived here only a few months, and have made so many friends already.