If your baby was born September the first thing I'd do is look on this forum for details of the autumn baby group (if August look for summer babies). There will be a whatsapp / facebook. The winter / spring / summer / autumn baby groups are started off on this forum, anyone can join and they are a great way of meeting other parents with babies the same age. I'm still the best of friends with other winter babies 2015 mums 3 years on.
In terms of good stuff to do with a 3 month old, check out...
Gymboree at Café G (google it)
Rhyme time at either Dulwich library or Peckham library - various slots
baby swimming? Loads of classes on at various warm hydrotherapy pools in the area
Monkey music, loads of people rate it
Heart beeps is a good baby sensory class, I think the version for little babies is called baby bells.
Postnatal yoga with Michelle is very baby centric, built around nursery rhymes etc for the babies: [www.michelemyoga.com
Some classes are friendlier than others, just totally depends on the people who attend on the day. Some get taken over by NCT groups who only want to talk to each other, but don't give up.
All that stuff should be sufficient to get you out and about and part you from your hard earned cash :-)
Welcome to the neighbourhood!