We saw HP and the Cursed Child in November and it was GREAT! Well done on tickets! There are tons of places close by that you can go to for dinner. We went to Carluccio's (Italian) which is a favorite of ours. We did make reservations though, as we were there on Saturday, even though our reservation was super early!
Are you seeing the shows on the same day or over two days? If it's same day, you won't have time to do much else that day as it's over 5+ hours just of performance time, with a two-ish hour break in between.
If you're museum-y type people, those are good without kids, as are Westminster or St. Paul's. Also, just wandering around is nice without kids going, "how much longer, my feet are tired, worst day EVER!" every 5 steps or so!