They are doing it because they have to (they are breaking the law if they don't), and it is not in your interests to ignore it.
You have a right to appoint a Party Wall surveyor at their expense who would work for you to ensure that the work was done in a way that did not undermine your foundations. If you just ignore it, they will just carry-on and although any future problems you experience (for example, cracking) are still their potential liability, you would have to take them to court to get redress.
Either go to CAB in the first instance, or phone a Chartered Surveyor and ask if about Party Wall surveys. CAB will probably suggest you do that anyway.
You will not have to have dealings directly with the neighbour, your surveyor will deal with it. The neighbour has a legal obligation to pay your surveyor costs.