If you can print your own boarding pass and through check luggage (or even manage without), your bigger danger will be keeping the kids amused for 40 minutes or so. Luckily at LCY, with its fantastic runway viewpoints and high chairs, this won't be difficult.

I have often done a 1 hr EDI to LCY to JFK transfer, not had a problem yet, apart from boredom.

There isn't a direct transfer route however, you do have to reclear security through main security. So out and then in again.