There'll be plenty to do on your doorstep, I promise.
Tiny Teddies at the Library on first and third Thursday of each month at 11am.
Storytime (under 5s)at the Library on Wednesdays at 2pm.
Check at the library to see what goes on in the hall in the same building (music class, I think).
For music, try www.kiddie-quavers.co.uk for pre-schooler music class.
"Songs and Stories" at All Saints Church 10am Wednesday and at Christ Church 10am Thursday morning. If you're not "religious" it doesn't matter!
Drop-in play session from 10am at the Baptist Church on Temple Sheen Road.
Nearer to East Sheen Primary School is Elim Pentecostal which has another drop-in session (not sure of day and time).
Palewell park swings; playground by Mortlake Station.
Lots of coffee shops where mums frequently meet which all appear very child-friendly.
Things like Push Babies in Richmond Park.
I would advise you to get nursery places sorted as soon as possible as they seem to get snapped up very quickly.
My boys are 16(!) and 6 so a lot of your age-group things are behind me now but I do know that Sheen is a very family-friendly place, with plenty to do without getting in the car, and you'll soon get to meet lots of people.
Also check out what's on offer in Mortlake just the other side of the railway.
Good luck with the move - whereabouts will you be?