If the conservatory has been in place for more than 20 years than the neighbour has acquired an easement for it to remain in place. Therefore the fact that it may be on your land is immaterial.
If it has been on your land for less than 20 years, and they won’t remove it, you can apply to court for an injunction to make them remove it although it’s more likely that the court would award you compensation rather than make them tear the whole structure down.
If the work has caused cracks or caused cracks caused the cracks to widen, then the contractor is liable for that damage.
It will be easy enough for a surveyor to look at the cracks and to give a report to say whether they are new or long-standing. Just because there are cracks, doesn’t necessarily mean that the whole thing is moving and will continue to move. It would need a structural engineer’s report.
However, their request for work to stop, adjacent to the conservatory, pending further investigations is not unreasonable and if you don’t agree to that, they could apply to court for an injunction and legal costs, and that injunction would be to make you stop work pending the conclusion of the investigations.
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