I work in the industry and have children so am very fussy about what I see / take them to! I wholeheartedly recommend Six - my son is younger than your two and absolutely loves it (it helped that he already knew all the songs off by heart before seeing it but you can listen to it on Spotify before you go and see if you all like the sound of it). It's modern, fresh, different, uplifting and a brilliant message about and for young women particularly. It also tours so you wouldn't necessarily have to see it in London.
Matilda is also fantastic and I would definitely recommend it. Six year old loves it and even my younger one sings most of the songs!
I actually haven't seen Frozen but have only heard great things and it has a brilliant team behind it.
The Lion King is good visually and one to tick off the list of big musicals if you want to but I'd put the other two above this personally.
Back to Future is amazing visually but I don't rate it myself. (Many others do, obviously, I just didn't really enjoy it and wasn't invested in any of the characters). Wouldn't jump out at me as THE one to go to for you.
The Play That Goes Wrong is a farce - goes down well if that's your kind of thing but I wouldn't put it up there for that age bracket.
As others have said, &Juliet is closing in london but would be a good one for them when in tours. All the songs are ones written or produced by Max Martin, so lots of Britney, Katy Perry, Backstreet Boys etc. Can also listen to the soundtrack on Spotify.
I second looking on Seat View to find seats that are affordable but not behind a pillar. Enjoy!