On Sunday there I took my missus, my 3 year old daughter and her friend along to Climb Newcastle in Byker. (As recommended by ti_pin_man - cheers!)
We did a Family Foundation session, which was membership and hire + an introduction to the centre, run through of basic safety, a little bit of basic coached climbing then climb on for as long as you like.
(Mini-S on the main wall with her instructor)
Nice place. The staff were great with the kids (and didn't mind at all that we'd brought a baby along too). Both kids and the missus had a good time and I was able to leave them in the cafe and do a bit more myself.
Good variety of climbs too (all bouldering). It is an old swimming pool, so there are shorter walls on the sides and bigger walls in the "pool". They have them organised into circuits so you can do a full circuit of the 14 yellow climbs (Grade Fb 3-4) then progress on to the black climbs (4-6).
Kids have their own mini walls upstairs and they run a "RockStars" kids club so you can leave them for a bit and get some climbing on your own.
Recommended to the OP (and anyone else in the area).