The Party Wall notice engages the Party Wall Act and once issued, you need to follow through, possibly instructing a surveyor and perhaps also paying for the adjoining owners surveyors surveyor. You will need plans and sections, together with Risk Assessments and Method Statements from the piling contractor. Allow 2-3 months.
If your neighbour does not give their written consent to the proposal, they will be deemed to be in dispute and 2 surveyors will normally be needed to draw up a Party Wall Award - all paid for by you.
If you chose not to issue the notice and just carry on, you run the risk that the neighbour could issue an injunction preventing you from continuing, but they would need to do this at their own risk and expense. If their injunction is successful, you would be liable for their legal costs as well as any damages. If they do not issue an injunction there is nothing else that can be done.