How long has the neighbours conservatory been in place for?
You say that the conservatory is cracked but you have no proof. Do you mean the neighbours are alleging that you have caused the cracks and that there were no cracks there before?
Is it likely that any of this work has caused those cracks?
How close to their conservatory have you been excavating?
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.
Can I clarify anything for you?
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The legislation governing this, the Prescription Act goes back to the mid-19th century. It is the fact that it’s been on the land for so long which is problematical for you. The idea of the legislation is to stop landowners standing by, watching neighbours build on land which doesn’t belong to them (trespassing) and then expecting the structure to be removed. Even if it had only been there for a few years, and there was no easement, the court are unlikely to order it to be taken down.
I think that letter is absolutely fine subject to a few typos.
It’s non-aggressive and it offers to put the work in abeyance near to the conservatory and to talk about it.
The suggestion about not doing it by text is excellent but I would suggest that you actually put down about talking face-to-face over a cup of tea