Kate - I also have a toddler in Teddy. Here are a few that have stuck in my mind, as well as the Tues/Fri playgroups at Bap Church, which I highly recommend...
There is a drop in soft play type place at the sports Pavillion in Udney Park Rd, think Mon-Fri mornings (small fee)
I have heard good things about Little Fishes at St Francis (?) church at far (Hampton Hill) end of Princes Rd, but don't know days
Highly recommend Diddi Dance at Salvation Army hall in Church Rd on Mondays (2 sessions, £6 pw). My DD's favourite!
There is a paid for Crafty Tales weekly (Fri am, £10 but good value) with dressing up, interactive story telling, singing and bubbles etc, and a craft (sticking!) activity - all on a weekly theme.
There is catch-a-balls at Bap church (poss also Tue and Fri?) Not been but friends do
Library does (free I think) story telling - you used to have to sign up in advance
There is Monkey Music at Landmark Centre. Again not been but recommended by friends
Further afield we love Jellybugs in Kingston - easy by train or bus, in basement of rotunda. It's a mini riot, with tea and cake for sale!
TW11 magazine, a new venture, has lots more listings, including a drop in play/meet at the church in Station Rd every week (Mons?). Not sure how you get a copy, but try some local shops and cafes, eg Astoria (coffee one opp Tesco). Or Tweet @TeddingtonTown, if you do Twitter.
Have fun, and maybe bump into you one day!