I find that the NT try and make most of their properties child friendly. Nostell Priory is local to us, in Wakefield, and we visited loads when the dc were little. It has a fantastic playground, loads of running around space, often has kids activities on and has a doll's house of itself, and some little bike routes. And a pizza van/ice cream van aswell as a cafe.,
Clumber Park, Notts, has an amazing playground and bike rides, huge estate, but no property.
Quarry Bank near Manchester airport is an old mill, "big house", workers accommodation - lots of interactive stuff and the usual high quality playground and cafe in the grounds.
Cragside in Northumberland is an amazing arts & crafts house, with lots of interesting stuff in the house as well as trails etc in the grounds - first domestic property to be lit by electricity,
In the SW we enjoyed Cotehele house and garden, and the mill and quayside that are an easy walk from there. Also Stourhead (all year round), Lyte's Cary and Alfred's Tower - a tall folly lookout tower on the boundary of Somerset/wiltshire/dorset. It's a good climb for great views, and lots of woodland to explore (the woodland is not NT, the NT bit is just the tower, not visitor centre, toilets, cafe etc, not worth the visit by itself, but good in combo with Stourhead).
Calke Abbey is Derbyshire is also great, big old family home, old toys etc. large outdoor space.
You could do a great holiday to the Lake District based on NT places - Wray Castle, Sizergh Castke, Allan Banks and more - great walks (NT land with free parking to NT members) at Aira Force (massive waterfall), Tarn Hows and Felllfoot.
A holiday my dc (12, 15) still take about was when we got the ferry to Belfast and toured the Antrim coast, taking in the Giant's Causeway and Carrick-a-Rede rope bridge (as NT sites, amongst others).
NT membership is also very handy for parking in the Lake District and Peak District, where they own and manage a lot of land.