Might be worth having a search for local woods and find out who owns them and asking them....a lot will say NO but if you're polite and ask enough landowners maybe one will let you.... or offer some of your labour in return maybe...or even to pay for the use.
Might also be worth asking the local Scout Troop leader where they go to do it...
Or look up some of the more rural campsites and either phone or better still pop in and speak to the warden....
Now that bushcrafting is getting more understood thanks to show's like RM's....they shouldn't look at you like your mad and so long as you pay your way and keep the place tidy they may be happy for you to use their campsite...
Hope that's of use to you....
I do mine on Charity owned land, I started off writing to the Trusties asking for permission to collect wood for my wood burning stove and as they saw me clearing the pathways and bridalways (I was clearing fallen trees to use and veggitation to get my log trolly through) they were well pleased as it saved them relying on volunteers to keep the paths clear and from there I asked if they minded me burning off some of the smaller off cut bowes from snedding the tree limbs...they knew me by then and were happy I wouldn't burn the whole place down and so said yes....then a while later I asked if they minded me camping out from time to time...again because they know me they said it was fine....although they thought I was mad wanting too! lol
What's that expression...."Softly, softly, catchy monkey"....seems apt here