Hello I am a solicitor with over 25 years experience. I will try to help you with this.
Were you in occupation at the time the extension was built and if so did your neighbour serve any notices on you prior to building the extension? when was the extension built?
I am a working solicitor and fit in helping on this site around other work. I therefore do not always respond immediately to your further questions, but I will respond as soon as I can
When was it built?
Hello. I am back now having been busy over the weekend and yesterday. Will answer you later today.
So your neighbour has extended his house about 13yeasr ago and the guttering extends over your land. He had no permission to do this. Damage has been caused to your land and now it is in the way of work you wish to do on your own land. The neighbour has an unhelpful attitude. You wrote to him 2-3 years ago. I do not know what was in the letter but I assume you complained about him extending over your land and complained about the damage he caused and requested him to stop invading your property and stop the leaks.
Your neighbour should have sought agreement from you before building across the boundaries of the properties and the general position is that he could not legally build this without your permission. Follow this link and go to page 16 to see more details of the procedure he should have followed.
You neighbour had no right to extend his property over your land he has not obtained rights to do so by adverse possession or by prescription. To the extent that this structure overhangs your land he has no legal right to do this.
What to do about his is a matter for you but you may think a polite letter pointing out the above and requesting he alter the drainage on his roof to remove the overhang would be a good place to start. It may be that stating that you wish to extend yourself, like he has, and hope to do it all with agreement etc. You may be in the legal right here but it would cost a great deal of money to enforce your rights in the courts and sorting things out by agreement would be very preferable.
Also do not simply remove his gutter as if you do so you may be responsible for any damage that is caused and you never know he may call the police and accuse you of criminal damage.