I have lived in London for most of my adult life, and my 2 kids (aged 11 and 13) are Londoners, but I’m not from London, so these are some tips from a local but with some non Londoner perspective.
totally agree with suggestion to not over schedule. Getting around London though it’s easy to do can be exhausting at times especially when busy. Based on experience hosting family from more rural parts of England, I think non locals are more likely to find this, so leave some time for rest, generally wandering etc. and I’d also not do a show every night - you may want a more chill evening away from the west end!
from Camden, a really nice and slightly more chill activity would be to walk along the regents canal, could go west towards Maida Vale which goes through Regent’s Park and you can see some of the zoo animals from the canal path, or east towards kings x and angel. The area around kings x as got really nice over the last few years, and I’d recommend it for a far calmer shopping experience than the west end
consider what sort of food you want to seek out and search for some recommendations for the best x or y cuisine. You can get great food from all over the world in London, that’s one of the best things about the city. Some of my top recommendations are:
lahore kebab house and Tayyabs, both in Whitechapel area (Pakistani)
song Que (kingsland rd, near old street, Vietnamese)
four seasons (Cantonese, they have a branch in china town and one off Queensway between Hyde park and Paddington)
koya (Japanese udon noodle specialist, they have a branch in the city near mansion house and one in soho)
for theatre/shows, look into off west end venues including national theatre, south bank centre, old vic, plus smaller venues like kings head theatre in Islington
I’d def check out an art gallery - the permanent collection at Tate (Modern and Britain), the national gallery and portrait gallery are all free. They do get busy so go early and I think Tate Britain is probably my favourite plus it’s less busy
some of the smaller museums are great, and also less busy. My kids recently enjoyed the Handel/hendrix house, round the corner from Oxford Circus (it’s where Handel and jimi Hendrix both lived, centuries apart)
in terms of getting around, also agree to not use the tube map to judge distances - it’s a diagram rather than a map. I’d recommend downloading the city mapper app to navigate - it will also show the walking route and for shorter distances in the city centre walking can be quicker. I’d combine your trip to Greenwich with a Thames boat trip - you could get the boat there or back. Very nice way to travel