The only exempt areas are enclosed fields of crops or farm animals and campers must pitch tents well away from buildings or roads. In recent years, some smaller areas such as Loch Lomond and Trossachs National Park have introduced by-laws that restrict wild camping requiring you to have a permit/camp in designated areas.
In England and Wales, it's the opposite. There is no right to camp in the wild without the express permission of the landowner. There are some areas where wild camping is common. For example, in the Lake District and Snowdonia, there are large areas where camping is accepted, providing reasonable precautions are taken, and campers must be out of sight of all roads and human habitation. But it's still not a right.
Dartmoor is a little different because you have a de facto right to camp and the national park does encourage responsible wild camping. See this useful Dartmoor camping guide.
CampWild works with landowners, trusts and custodians or permitted wild camping.
The advice is to camp high up, pitch late, strike camp early, and leave no trace.