As long as they plant trees that have had a few years in a nursery first they should be able to coexist despite the roots. Heather roots are quite shallow, tree roots would generally be able to penetrate deeper underneath them. Eventually the canopy of the trees would kill off the heather allowing other plants to come through.

The biggest issue would be keeping the deer numbers low enough to allow the trees to establish and become mature.

That's one of the reasons why there's been a push to reintroduce wolves and lynx, to keep deer numbers manageable because decades of hunting and culls haven't been working.