Deep Sea World is not very good, don't waste your money. And it's a bit of a drive. Although you could get the train from Edinburgh and it goes across the Forth Rail Bridge which the kids might find exciting? It's only maybe half an hour on the train.
National Museum is good but much less interactive than it used to be, like most places nowadays sadly. Dynamic Earth also used to be very interactive, I don't know what it's like now.
Camera Obscura is fab but lots of stairs and no lift. You used to be able to borrow baby carriers, don't know if you can now.
You can push a pushchair to the top of Calton Hill and you can see lots from there. There's a wee coffee hut at the top for a snack once you get there. There's a small museum about the observatory and you can climb up the tower, maybe your older one could do that?
Take the tram from the city centre to the Krispy Kreme doughnut shop, watch them get made, eat one and get the team home? (My kids like weird stuff like watching the tram floor move as it goes round corners and watching rows and rows of doughnuts get fried).
There's a new shopping centre just opened at the bottom of Calton Hill called the St James Quarter, it's got a lego shop in it and also a leisure sort of place with pool tables and bowling if that would interest your older one.
Portobello is lovely, long sandy beach and promenade, easy to push pushchairs along. There are loads of cafes, ice cream shops, amusement arcade and toilets. You could easily pass a sunny afternoon there. There's a great takeaway pizza place not long opened, you can get a slice of pizza and chips. Nice independent bookshop and other wee gift shops on the High Street just 5 minutes walk from the beach if you get an hour to yourself.
Cramond beach, you can walk along the causeway to Cramond Island, don't know how easy that is with a pushchair though, a couplenof ice creams dotted along.
You can walk for miles on the flat from Cramond along to the beach cafe at Silverknowes, I can't remember what its called but you can't miss it.