If you do fancy it, I have been to the playgroup at the Forest Hill Methodist church with my baby (5 months) that Derbyhilltop mentions, and it is very nice - there is a quiet peaceful room just for the babies, with playmats on the floor (so you don't have to have the babies on your lap, or take your own playmat, as you would in a pub/cafe), there is real coffee!, tea and biscuits. It's £2, so cheaper than buying coffee elsewhere.

The toddler room does get busy, but I think the baby room is a bit of a hidden treasure!