Going over the article, it mentions three constraints pertinent to obtaining timber, and it seems that their intersection is empty.
The constraints for obtaining timber are:
1. A UK source is preferred (due to national pride?)
2. Price needs to be very cheap given the archeologists' non-profit budget
3. Veteran trees are necessary for a reconstruction, but specifically these are a scarce and precious resource.
It seems likely that either they relax (1) or get funding/sponsorship to relax (2). (3) does not admit relaxation