Non25, if you're still listening in...
Horniman is a fantastic resource for wet days, loads to look at and do - my ds particularly enjoys losing it in the interactive music room.
If you're in Crystal Palace/Upper Norwood (like me) then check out the library on Westow Hill - good children's section and the staff are most tolerant. It has a Wriggle and Rhyme session on Tuesdays (I work part time so I've never been to the session).
Crystal Palace park has good bits and bad bits. It is enormous. Bit awkward for buggy access from CP Parade. The actual play area is small and not great for toddlers, but the dinosaur lake, maze and surrounding areas are lovely to wander about in. There's a toddler gym in the CP sports centre, but you have to pay for a whole term's courses. I took ds to a freebie intro session and he loved it. There's a handy park hidden behind Morrison's/garden centre (Westow Park?) which has better equipment. Dulwich Park wins hands down on the equipment front.
Also in CP - there's a handy (and very good) aquarium shop on Westow Street for free fish-gazing. Also a good book shop a few doors down which has a little kiddy area. There are several playgroups in CP, none of which I've been to, but my ex-childminder really rated the one on Gipsy Hill, in the old church.
Don't know about Dulwich but I think there's quite a few mumsnetters out in East Dulwich, so I'm sure they can advise.
Good luck!