Hi Daniel - Hamsterley Forest is not that far west of Durham, but is not really big enough to accommodate a 20 mile walk.
I would look northwards - if you are determined to be in amongst the trees then surely Kielder beckons - miles upon miles of easy forest roads , trees by the million and I'm sure plenty of decent hidden places to camp. Midges might be a problem, though, and the Forestry Commission might take a dim view.
Better in my opinion would be to head towards the Cheviot, I'm sure that you could string a good route together there, and away from the Pennine Way and The Cheviot itself, it is likely to be very quiet. I've not walked there myself (on my 'to do in the near future' list, but the country to the south looks particularly interesting, some steep convoluted river valleys dissecting the uplands.
Finally, on the subject of wild camping:
Arrive late,
Leave early,
Be discreet,
Leave nothing
PS - just over the border in Scotland it is legal to wild camp (in non Covid times, that is)