Don’t lose sight of the fact that you stated upthread in reality you would not have objected to the planning application.
In the cold light of day is it more that you are feeling railroaded by this neighbour?
Contesting all of this is quite big and you need to be sure of your motives. Ie are you just shocked and overwhelmed with everything and then being asked to sign a party wall agreement that you don’t understand.? Or are you adamant you would never have purchased the house If the extension was in place and actually built?
Ignore the disruption as you could have been in the new house and next month a planning application could have been submitted and that’s life so to speak.
You need to be really clear on these points as once considered you may feel differently. It may be this isn’t your dream property and more of a compromise which is affecting how you feel.
Whatever. I do think new neighbours are being extremely pushy and exploiting the situation. You don’t actually own the property yet and pushing for decisions on party wall agreement are premature at best.
This may well all work out fine. But the party wall agreement is to protect both neighbour and you. You should ask for your own independent party wall surveyor- most people do and you are not being difficult in insisting on this.
Yes of course the building works will be annoying but would you still have wanted to buy the house if they were completed? Of course you don’t have to agree to the access they are wanting. That’s just making the build cheaper for them. It can all be done on their side even if that means craning things over the top.
Keep calm and consider things logically.