Upvote:2
How likely am I to actually make the connecting flight to London?
I'd say 80%+. If everything works as it should, you should be fine. Things can always go wrong, but that's hard to predict:
Assuming this is Eurowings, missing it is not much of a problem. They will just put you on the next flight to London and on most dates there are two more the same day after the one you missed.
The additional complexity is that I will be travelling with 3 children aged between 7 and 9 (i.e. one adult with 3 children).
If you feel you need extra time, you shouldn't book a tight connection. This being said, Stuttgart is a small airport and 7-9 are typically don't require a lot of extra time. Of course, that depends a lot on the personalities and behavioral patterns of the specific kids involved.
If you want to be extra careful: chase them all to the bathroom in the plane about 1 hour or so before landing in Stuttgart. This may save you a pit stop on the ground.