Every year there is an open day/weekend for a lot of the sites which unless you are a National Trust member you cannot get into for free.
The link should take you to the page where you can look at some highlights they've picked out for you, but check other local places as it will be a free day out and it may even be sunny enough for a picnic.
Here is their spiel:
Over 4,500 free events will be taking place across England's regions for this year's Heritage Open Days, 6th-9th September. We have picked for you some of the best, region by region