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The British Constitution, extending all the way back to ancient authorities like the Magna Carta, informs us that all land in England and Wales is owned, and hence you need the owner's permission to camp out.
Having said that, the rambling constituency in England and Wales is quite powerful, and gets outspoken support from the media. Parliament has acknowledged this with several acts protecting the right to roam and ramble, most of which are derived from the original 1949 Act. Once enacted, Parliament generally hands countryside access over to the Byways and Bridleways Trust for on-going administration. The rest is delegated to the local authorities. So there is no single answer to your question.
As far as 'wild camping' goes, Wales is known to be particularly relaxed and will tolerate campers who keep a low profile and do not create a disruption.
You asked about the Lake District, this region is pretty much the same as Wales. If you camp on a hill well away from a road or byway, keeping a reasonably low profile and do not create a disruption, nobody will object. The Lake District National Park has published guidelines for wild camping.
You also asked about Cornwall. In Dartmoor, camping is permitted for a maximum of two nights as long as you are not within 100 meters of a public road or enclosed area. Of immediate interest is their interactive camping map.
As a general rule, if you have any doubts, the relevant local authority is the best place to check.