Everyone who has suggested Lanhydrock... We love the sound of it. Can you spend the day there? Is there anything else near by worth seeing when we head over?
Easily! There are bike trails there but in my experience even the easiest is too tricky if they're not really proficient cyclists yet - not very long but lots of bumps and curves and hilly bits to negotiate. But if you have older children who're confident cyclists and you're bringing bikes they'll probably enjoy it!
There's a lovely play area there. Will be manic this time of year of course, but they can still kill an hour there no problem. It's right next to the cafe so you can collapse with a coffee while they do their thing.
The house is a little walk away, which is usually fine but for lazy-bones' is a bit of a trial coming back. The house is pretty awesome though I think. They have real rabbits and pheasants hanging up in the kitchens which upset me as a child, so be aware of that if your kids are especially sensitive. Mind you it's full of animal heads and skins anyway, typically Victorian/Edwardian. If you ask for it they'll give the children a 'things to spot' sheet.
You can go on long walks in the grounds if that's your thing, and there's a stream with some great places for children to have a good paddle and play, so consider water shoes.
If you're coming quite far into Cornwall, there's a NT gardens called 'Glendurgan', I think on the south coast the other side of Falmouth, which is pretty cool. They have a maze and a giant-stride. You might even be able to get down to the beach there but I may be thinking of Trebah which is next door.