Teddington School has been Good for a while now, and getting back to its previous popularity. 10 years ago it was heavily oversubsubscribed, and it is now starting to return to that position. (It had some financial issues, which have been addressed by it becoming part of a Multi Academy Trust, and it has had two new Heads since it got its RI judgement - the first one was credited with turning it around, but then moved to Orleans Park. The most recently appointed Head was previously a popular deputy at another local school).
A lot of families in north Teddington, Hampton Hill and West Twickenham send their children to Turing House School, as it has its main admissions point in Fulwell on the Twickenham/Teddington border.
Hampton families use Hampton High (but watch out, as some southern areas of Hampton have been out of its catchment in the recent bulge years).
West Twickenham property prices are pushed up by proximity to Waldegrave (girls school) but families there with boys or who don't want all-girls use Turing House, Twickenham School or Richmond Upon Thames School. The South Twickenham families that are closest to central Twickenham can still get places at Orleans Park, but only just.
Hampton Hill is my favourite area - a bit cheaper than Teddington and Twickenham, and fewer trains (from Fulwell Station) but on the doorstep of beautiful Bushy Park, and with the fantastic Hampton Open Air Pool at one end of the High Street.