any ideas or suggestions

Victoria park was a nice place, i taught round there in the early 70s!

The Museum of Childhood is in Bethnall Green, and sounds a lovely place.
However most of the activities seem to be for children of 3 and above and there's possibly a good reason for this.
I think most children of two would be quite tired after being at nursery and thrilled to bits just to see their Grandma and Grandad and have a few stories or games at home, or in good weather just go to the local playground!
Trips out are nice, but at this age better kept as occasional treats imho.
Google activities for children in East London -quite a lot comes up.

Hello sweatpea my sister lives not far from there and there is a swimming Baths at Mile End which is fairly new.

I have never come across an art gallery that would even remotely entertain a 2 year old!

I suppose you could look at a Stubbs paintings and say "isn't that a lovely horse" or perhaps Dali "Look, the silly man has tried to make a telephone out of a lobster!" Munch? "Oh look, that lady seems to be upset, do you think she has lost her Oyster Card?" Pollock "Just look at the mess he has made! I expect his mummy will be very cross, don't you?"

The V&A Museum of childhood near Bethnal Green Tube station.
That would probably do young children. Google it.

You can usually find whats on offer for young children in local libraries (if still open)

If you go to the library you will be able to find out what is on in that area. Perhaps there is a messy play session or something like tumble tots? My GS loved the messy play and also loved Tate Modern, he is 2 3/4. He has been to the Natural History Museum twice and loved it both times. In Wimbledon there is a play cafe, perhaps there is one near you too?

My 3-year-old GS loves the Discover Centre, 5 minutes from Stratford Station. www.discover.org.uk/.