Bayonne Nursery/Childrens Centre Tuesdays 9:30am - in the children's centre attached to the nursery. Great staff, small outside area, lots to explore, fruit time (I think they still do that). No charge.
Twynholm Baptist Church Tuesdays 10-12 - indoor only, friendly and plenty of tea, coffee, cakes and biscuits. Healthy snack served up for the little ones. Song time at the end. Charge £2 per family.
Dorset Wharf Playgroup, Rainville Road Monday, Wednesday, Thursday 10-12 - overlooking the river, indoor and outdoor areas, crafts and toys, snacks, tea & coffee for mums, lots to play with, song time. Charge £2.50 per family.
Bishops Park Playhouse, Fulham Palace Monday - Friday, 9:30-?, possibly afternoon session too - Brilliant setting, especially as they get older. Huge garden, sandpit, gardening sessions, water play, lovely things to play with inside. Lots to explore. You'd need to find out the times though, as I can't remember. There is a charge similar to the other, but I can't remember what that is either.
Queen's Manor Primary School do play sessions for little ones and music on a Thursday. You'd have to call for the details of that though, as I've not been in a couple of years. St Andrew's, St Thomas, and All Saints churches all also have playgroups, though I've not been to them. St Paul's in Hammersmith has one in their new building too.