mamatwo, hear what you are saying and know 100% what your going through - - I had same problem.
1st - It is right to say that the 'hedge' it'self dose not define the bounardy line, only your 'Deeds' will show that.
2nd - Whoever planted the hedge, is the owner and can do with it what they which. I.e., if the owner before you planted it, then it is of course yours now.
3rd. - The previous tenant who always cut it, may have only done so because they 'Liked doing it', but that never made it theirs per se.
4th - If You are happy for the Council to remove it, or whatever they want to do, then let them.
However, you must consider this, and it is not the fact that you are the 'Fool' in the matter, - by no means! BUT
5th - If you feel, for the sake of the aesthetics, of your own arear / outlook, then cut back, whatever is growing on your side. & even go that bit bolder and cut it down to the size you would like it to be!
The 5th option is entirely yours, and if you take this one, then no-one can make you put it all back!! Furthermore, by the time they all argue the toss among themselves, you are looking at an eye-sore! Right??
You will never be the 'Fool' if you are simply sorting out your own aspect! Good Luck in whatever you decide! Even if you cut it back yourself and hit the neighbour & Council with YOUR Bill for doing the work!!
Ask all parties concerned the following question: "Should there be a storm, and this 'hedging' damaged property, then who is responsible for the repair bill!? " I think you know what the answer will be!! - - - Won't be the Council for a start!