Building a single storey rear extension that is 25 sq.m. that satisfies all requirements for exempted development.
If the next door neighbours object, is there anything that they can do legally to halt / prevent construction proceeding smoothly once the builders have started?
Your neightbours can’t object where no planning application is being considered, which I presume is the case here.
They can contact Planning Enforcement however if your development is exempted and complies which the Regulations in all material respects, there is no case for enforcement.
You could consider applying for a Section 25 certificate from the Planning Authority for peace of mind. This will confirm the exempted status of your development.