You say they had people around with clipboards taking photos? That could be insurance assessors, for their claim on their insurance for damage.
Go around/get your dh to go around and take a photo of the water damage, it may help indicate where the problem might be on your side if it is on your side.
Check the water meter overnight as suggested. If the water meter hasnt done anything funky overnight I would insist they check all other possible options first, before you cause any damage to your property. I would also go into your loft and check there is not water coming in via your roof etc.
If your water meter does change overnight then you need to find the cause of the leak as it will be damaging both properties. Star with anywhere the pipes are easily accessible e.g. in the loft, and if you cant find anything then call a plumber.
You dont need to pay for their damage. So if you havent got buildings insurance the most you have to pay is for the plumber to investigate and repair the leak. You will need to pay at some point to also make do any damage caused to your house by the leak or cause when finding the leak, but unless its rotted your joists etc you may be able to live with it until you can afford to fix it.
Depending on the issue i would expect the plumber to be hundreds not thousands
If you havent got buildings insurance get it, not having it is a false economy, what if you get storm damage, more burst pipes etc