There are good options for "mountains" in the Peak District that don't have the height, but do have that atmosphere. Bonus points for well laid tourist paths.
Dovedale/ Thorpe Cloud
Mam Tor
Surprise View (a ridge with lots of lovely rock formations, car park near the top and good for a beginners wander and explore.
Mine (6&9) have done the 5 mile loop around Mam Tor a few times. They love the collapsed road section on the way down which means it's just the last flat bit where the whinging kicks in.
Dovedale is within an hour from us so we can romp around there regularly in the summer months which is good for building stamina.
From London, it's worth considering more accessible upland areas like the Chilterns to test the ground.
I'd consider Kinder Scout in good summer weather, but am now rusty about the details at the top, so would want a good day even without it's tendency to inclemment weather. I know my DCs haven't been out enough in recent months to have the stamina at present.
Lots of sweets. Spare warm layers. Walking recently, I had a survival shelter and sleeping bag for any delays/ mishaps. DS1 is the kind of child in need of a good stick, so brought one he foraged earlier! On this occasion, I was confident that we knew our Mam Tor route well enough to not need a map as we've been up a lot and it's fresh in mind.
We did the Cat Bells in the Lake District when they were 5 (walked all) and largely carried our 3 year old.