It depends what days you're travelling and how old the family is, so check opening hours, but:
- Tyntesfield (mentioned, near Bristol) is nice; can get very busy
- Knightshayes is lovely and big (Devon)
- Lanhydrock (gorgeous, gorgeous gardens) - I see a few people have mentioned this one
If you don't mind a little bit of a drive from the motorway (under 10 minutes), then Clevedon seafront is not a bad place to stop and stretch your legs usually a lot quieter than Weston-super-Mare, where I live. There is a small NT property, Clevedon Court, in the town, but you really do need to check the opening days & times for that one.
I'd suggest Brean Down or Sand Bay as well but I think they might be a bit more of a drive from the motorway than you'd want on such a long journey. (Yes, I have done it, several times. ;)
I don't know if the Gardeners' World 2 adults, pay for 1 ticket is still available but that has masses of gorgeous gardens down this way.
And my favourite place to stop on the way down to Cornwall is the Orange Elephant at Taverners Farm at Kennford - again, a little bit of a drive from the motorway but not too long a nice stop and they do really lovely ice creams and coffees (only open Fridays, Saturdays and Sundays though).
Oh, and there is also the farm shop at Gloucester Services now - not quite as good as Tebay and a bit more expensive, but a good half way option.