If your neighbours don't agree (or fail to consent), you are formally in dispute, and only appointed surveyors can produce party wall awards (the document that resolves disputes if your neighbour doesn't agree to the work).
Under the Party Wall Act, the procedure for dispute resolution applies in two situations:
Prior to work beginning: when the Building Owner and any Adjoining Owners do not agree on the works to be executed, or the Adjoining Owner fails to consent to them.
After works have started: when an Adjoining Owner suffers loss or damage to their property due to the works, even if they consented to them.
When a dispute comes up under the Party Wall Act, the parties involved are obliged to follow the procedure set out in the Act, and appoint a surveyor.
Once the surveyor is appointed the Act does not allow you to remove them and appoint someone else, so it's worth doing your homework on them.
As with all bureaucracy, and especially with architecture, it can take a significant amount of time to get details checked and agreed between all parties. Starting early can save you a lot of cost and effort.