I'm originally from Milton Keynes so just over the border from Bucks there is Salcey forest, Whittlebury Woods there must be literally dozens.
Here is a link from google which may help your search.
I'm in Northampton, we have little wooded areas (there's quite a big one in Lings just 2 mins off the A45, 5 minutes down from Wellingborough Mcdonalds roundabout) but privacy would be a problem especially outside of school hours because it's right next to a housing estate.
Personally as CB said Salcey forest is probably your best bet but again it gets pretty busy on nice days.
There's a chunk of wooded spaces north of Northampton - come off one junction up from Northampton on the M1 and hang a right and have a drive around is my advice. I've shot nudes right on the edge of a main road (model got a bit carried away!) and didn't get too much trouble. I've certainly shot lingerie in the woods up there as well.