I'll just leave my thoughts from a non-builders perspective.
I was in exactly the same boat, in some extent, to you. However, I had no issue with my neighbours building up to the boundary.
It is their land, and if the planners agree, it is up to them to do what they want. It was a single story extension down the side of our properties. I had a 1 metre wide path they had a 2.3m drive.
But, what I did have an issue with is what they did up to and on the boundary. It seemed that my neighbours were unaware, that the guttering had to be taken into account as part of the wall. Despite telling them this months in advance the builders either hoped I would forget or ignored me. The gutters and eaves should not overhang the boundary.
So the wall, in effect, has to be set back enough to compensate for this. The foundations also should not really go over the boundary line, but as others have said, there are ways to do the foundations that can avoid this anyway. But, if the wall is set back because of the eaves and gutters, the foundations should be OK.
My neighbours ended up reducing the eaves and guttering to the minimum possible. around 15cm. It was a bit of a bodge on their part, but it was up the boundary to the nearest mm.
All I was bothered about is that the boundary line was agreed and that it was not crossed. The boundary was delineated by the edge of their driveway (it was part earth on my side, so no boundary markings). Without accurate measurements, which their builders were not forthcoming with, once this was removed, all visual evidence of the boundary would also disappear. So accurately measuring where the boundary was, was very important.
It was difficult at times as it put a strain on our relationship. The problem was that the builders seemed to hope that they would be able to start the build before I noticed there was an issue, at which point, it would have been too late to do anything.
A party wall notice can apply in this instance, Line of Junction notice. This forces, in some ways, their builders to give details as to what they plan to do on the boundary. I used it as a last resort. It is a flawed piece of legislation but can be the only protection you have.
Your neighbours should be submitting a line of junction notice of building up to or astride the boundary. If they haven't, again, their builders may be relaying on your ignorance, and avoiding doing it.
If they submit plans which show a set-back, taking into account the guttering, you can agree.
If you disagree, a party wall survey is called upon. You can either get a single surveyor to act as an independent third party.
You can also take the extreme step of having your own surveyor each, this is usually overkill on such a simple boundary dispute and the cost can be extortionate.
The person doing the building usually has to bear the cost too.
In the end, asking for the requisite notices, which my neighbours builders had failed to do, sharpened their minds a bit, and after a few tense weeks it turned out OK.
I let the builder put scaffolding on my side. This ensured it was easier for them, They did not have to build overhand so it looked a lot neater. They had to walk through my garden to get to the rear of their property. I had no issue with this even if it did look a mess at times.
As above, it is all about compromise.
They are allowed to build up to the boundary. The only thing you can do is ensure they respect where the boundary is and build in accordance with the regulations.