They definitely can not cut your tree without your permission. The tree cutters are trespassing and should not have entered your property without permission.
We had a similar thing happen in the place we were house sitting. We arrived back and found someone cutting the tree on the property line. The tree cutter tried to claim that the branches were overhanging and dangerous, which is one case where they are allowed to cut them. Even though they tried to make excuses, they cleaned up and got off the property fast when we threatened them with trespass charges.
If you need some legal advice, the Citizens Advice Bureaus are great. They would be able to tell you what your options are. And the best part is that they are free.