Boundary Dispute

i recently moved into a new house and since i moved my neighbours have built a fence inside my boundary and are attempting to claim that this is their land. I have ownership of the alleyway around my house and the neighbours are located on the other side of the alleyway. The neighbours boundary is nowhere near to the location of the fence, but there is a grey area between their boundary and my boundary. 

The fence was built without any input from me and the fence posts are located completely on my side, which i believe is incorrect. They claim they don't want to dig through the paving slabs in their garden....

We had a discussion after the fence was built and we came to a verbal agreement that the fence posts would be moved by their builder inside their garden, and the shape of the fence would be changed to match the grey area between our boundaries. 

3 months passed where no work happened, and i managed to meet with them again, in which time they completely went against everything we had discussed previously. They also claimed they had the right to do this, because " they had been there 40 years, I've been there 2 minutes".

Alongside this, i have a garage (built by the previous owner) which does slightly pass their boundary, this garage was built over 25 years ago. My neighbours also kept mentioning the garage is on their land, and if it moved they would take the land. For me, the garage will stay where it is, and i now own this land because the garage has been there so long. I don't believe this is my issue as they should have mentioned it when the garage was built.

Am i correct in what I'm saying? I am currently involving solicitors as they do not listen to anything i say and are willing to say anything to try to push me to agree.

Hi Daniel - You need to speak to the solicitors that you are involving.  We at HM Land Registry cannot give an opinion on such matters as that would be going beyond our remit.