Hi Laura - Fulham, mumsnet and motherhood - all good choices. Here is a whopper of a reply. I am not a massive poster on MN but when I get going I do like to go on !
I have lived in the area for a long time and still find it a bit boring some days even though my DS is 17 months and able to go to lots of the toddler things around most of which are not really a great option for you at the moment whilst your little one is so small but they grow fast.
What really changed our days for us was when I joined the gym at Virgin Active at Lillie Road. (I am not a natural gym goer) and started using the creche which is very very good value, takes babies from 3 months old and has wonderful staff. I did not join until DS was 1 and really wish I had joined earlier as they really take care of your baby, come and get you if they cry and you can start off leaving them for just half an hour at a time (they have cots there or you can leave them sleeping in the buggy in the creche). I then went and did half an hour on the cross trainer and finally did something about my baby weight (a year on!).
What is great about the creche is that you inevitably end up talking to other mums there and before you know it someone will ask you for a coffee or you could be brave and ask them. I think new mums in particular are very open to it as it is a difficult and isolating time if you are not surrounded by family and friends. Obviously signing up to a gym is a lot of money but you do an off peak package and I think you can even use the creche if you are not a member but would need to doubel check this
Here is a list of other things I know of in the area (I have added Putney because if you live that way in Fulham it is just as close to get to as some of the other things to do)
Here is a pretty random list which will be much more useful to you come September when your baby is sitting up but some things would be good options for you now......
Buggy Fit in Wandsworth Park Thursday 10am. Great way to trim up and meet new mums - esp with babies. You could probably google for details. I think this would be the best way given the age of your baby to meet other mums
Eddie Catz in Putney for a nice soft play are, feeding facilities and comfy sofas. Good for breastfeeding
Coffee and Crayons - Fulham Road, Fulham Palace End. Again good for breastfeeding
Rainbow Playhouse - in Bishops Park Playground from 10- 12 every day for £2
Swimming lessons at Fulham Pools (linked to Virgin Active) are a good way to meet people
There are always lots of things on at St Dionis Hall (opposite the White Horse in Parsons Green)
Mother and Baby group - 10 until 12. Think this is Tuesdays or Thursday
Amanda's Action kids - Tues and Wed 4-5pm
I have also seen French lessons there and Monkey Music. I suspect if you phoned the administrator for St Dionis Church she would have a list of who she hires the hall out to as there is no list in the hall and I only ever notice stuff on if I happen to be walking past
I think there is sometimes a Baby Massage class held in the basement of Wow Baby on New Kings Road
There are also playgroups at
St Marys Church the other side of Putney Bridge Tuesday 10 - 11pm
All Saints on Fulham high street - Tuesday 10 - 11pm
Sands End Playhouse
You will also be able to do Gymboree in Putney Exchange and
Little Acorns music group in St Marys, Putney
I see lots of new mothers at Sallly's cafe in St Marys in Putney and there are always loads in Bishops Park playground, pool and sandpit and also Fulham Palace - the lawn is FULL of mums and children on a sunny day - no dogs allowed which means you can let them crawl/run around without fear of poo!
Look out for the free magazine Angels and Urchins you can pick it up at Virgin creche and Coffee and Crayons which has a huge list of things to do in the back
Hope all of this helps !