20 minutes should be plenty if you know where you are going, don't dawdle, and don't stop to buy crap at all the stores that will be on your path. Get out of the Shinkansen side into the JR side through the JR transfer gate (same ticket) and head towards the Marunouchi (West) side, North or Central exits.

Find the escalators or elevators that go down into the bowels of the Yokosuka-Sobu line. The NEX usually leaves from track 3 or 4 of the Yokosuka-Sobu line, at level B4F. It's all well-signposted, although you'll probably be doing 5-6 floors of vertical down from the Shinkansen tracks, which are outside and on the Yaesu (East) side of the station to the NEX tracks, which are on level B4F of the Marunouchi (West) side of the station.

You could probably run the gantlet in 5 minutes without luggage and without dawdling. Add time for the 4 kids + luggage, and 20 minutes should be fine, but no dawdling or bio breaks!