Sounds like a nice visit being planned!
Visit museums for their shops - like the Natural History Museum - most probably has animals!
and Science Museum and maybe they have other museums.
In Edinburgh there is a Toy Museum which has old lead animal figures.
Hamleys will have the regular toys in it, they have a huge store in Regent Street.
And maybe there will be a small selection in Harrods, in their toy department which is in Knightsbridge.
One thing of note, if you like bronze sculptures and stone relief, is a wonderful display of Animals in War Memorial by Hyde Park, East side.
It has life size horses and donkeys and is very moving. It's dedicated to the animals who have served in all the wars. Every time I go to London I make sure to revisit it.
and I found an article that lists toy shops in London. Depends where you are though as it's a big city.
And if you visit London Zoo, they will have a gift shop.
Also if you visit any Botanic gardens there, they should have gift shops. In Edinburgh they have one, so London probably has them too.