DD is nearly 11 and while we are SE London we often go and play tourist with friends from abroad.
Zoo - nice but if you have any decent one near you better take it off the list, nothing really special you can't find anywhere else.
There are children tours at the HoP, very good, 2-3 hours but very worth it. We went last year when DD just turned 10.
Buckingham Palace - they do the summer opening but I personally think if it is the first time London I would skip it.
HP World - do check if there are any tickets available, holidays and weekends book out months in advance.
Show - on Tuesday, 12th June, the Kids Week starts. If you go in August you can get a free kids ticket for each full price adult one. Google it and be flexible with shows and dates. It starts at 10am I think and lines will be VERY busy.
Museums - they are free but food is expensive. All have picnic areas and lockers so take food with you and save your cash for cakes.
Have a look at serviced appartments. I find an appartment easier and cheaper than a hotel if I go somewhere for more than 2 nights. You can get your own food, hotel breakfast is often expensive. You can treat yourself in lots of cafes for a nice one if you want.
There is an afternoon tea bus tour. You go on a refurbished routemaster and eat while having a drive. Not sure how much you can actually see but a bit more unique than a hotel one.
Greenwich - you can check the shows for the planetarium on their websites. The museum itself is free, the planetarium is up the hill, the Maritime Museum is near the river. You can easily spend a full day there, esp. if the weather is nice.
At 9 DD wasn't really interested in the playgrounds anymore. Princess Diana has a waiting queue if it is busy, we gave up after 30 minutes when nothing moved. I wouldn't count on getting in unless you go there first thing.
Windsor Castle or Hampton Court Palace are accessible by train but it is a long day. You can easily do ONE long day out but otherwise London means long time walking, queueing, it gets hot and people can get irritated by tourists.
There is absolutely nothing special at Hamleys apart from the fact that you can get everything cheaper elsewhere. Oxford Street is a mess, chain stores you get everywhere. If your children like Lego there is now a huge shop at Leicester Square where you also find a M&M store.
London is an amazing city but be prepared for not being able to see everything and be very tired at the end.