Firstly, you need to settle this without legal proceedings, as otherwise the only winners are the solicitors
What amount is this guy talking about 10mm, 100mm ? How does he know the wall has been moved?
Unless the walls are dimensioned on the deeds or other documents, then there is no definitive boundary line. The red line on the land registry plan will be about 150mm wide in real terms when scaled up.
So the only way to define the boundary is for a surveyor to comment and give his opinion on where the boundary is most likely to be, taking into account the site layout. But even so, fences and walls may be moved over time and the fence or wall is not necessarily the boundary line
Although I don't suppose that you used the Party Wall Act when you did the recent work, you may be able to infer party [fence] wall status or use the terms and guidelines in this Act in any future legal argument. The PW Act wont help you now as such, but it can be used as persuasive argument.
If the neighbour has begun a civil action, then check with your house insurance policy, as you may be covered and they can deal with it for you.
If not, then you probably first want legal advice, then you want a surveyor who is experienced in boundary and party wall matters. You may be willing to appoint a single surveyor as joint 'expert' which will reduce costs, but if so, make sure he is truly independent.
But if he is taking you to court, then wait for him to state his case. He must prove his claim, and he must provide the evidence on which his claim is based.
If you do get a Solicitors letter, then don't be intimidated by its wording or content. It will be full of bluff and assumption, and written to make you think that what they say is correct.