Any landmarks, monuments or castles in London that I need to pre-book?

If you want to go to the Harry Potter studio tour you absolutely need to prebook - and do it right now.

Sky Garden, for the views, needs prebooking three weeks ahead even though it’s free.

Most of London’s major museums are free to enter so no prebooking needed.

Tower of London, turn up at opening time and buy your tix then head straight for the Crown Jewels to avoid the queues.

Buckingham Palace tends to beef prebooking but I’m not sure it’s open in April.

For any sights that require a ticket (so, not the free ones), I suggest prebooking simply because it enables you to skip the ticket-buying line. For some sights, like Leavsden Studios as mentioned, prebooking is also a necessity in order to ensure entry on your chosen date.

I highly recommend the Harry Potter Studio tour if your kids like Harry potter at all. Tickets sell out fast so definitely book those now.

We pre purchased our Tower of London tickets but there really was not advantage to doing so as we had to exchange out confirmation for actual tickets on site. My kids ( 12 and 9) really loved the Churchill War rooms and pre purchasing tickets for that one was a huge time saver.

We also pre purchased Hampton Court palace tickets. We loved it there

We looked into the city passes but based on what we wanted to do, it was not advantageous at all. I think you need to make list of what you want to see and do and then compare it to the cost of the city pass.

We bought our Oyster passes at the airport and it was such a time saver. Our 12 year was half price and our 9 year old was free

April is not peak tourist season, not quite, but the weekends in particular can still see long line-ups at prime attractions. The locals love to use the resources of their wonderful city too. It's true that major museums are free -- for the permanent collections. Special exhibitions that attract popular attention will usually charge a hefty admission. With three youngsters I can't imagine that art museums are high on your list anyhow (or theirs) but a stage musical or comedy could be a treat. Lots of useful info, including good/bad seating in particular theatres, here:
http://theatremonkey.com/
A cruise along the Thames might entertain kids but you can wait until you are there to see what's going on with the weather.

I’ve never pre-purchased any tickets for any attraction in London (also including Hampton Court, and Windsor castle) but I’ve never traveled there during high tourist times either. For trips outside of London I have purchased the National trust or Heritage pass if we intended on visiting a number of those locations.