I suspect it is highly unlikely there is anything wrong with your section, other than its backed up from the issue downstream. Chances are when that is cleared, your issue will also be sorted. Previous advice was assuming the Water Company employees had done their job properly, checked the run and that downstream chambers were clear. It now seems they hadn't...
Might be worthwhile taking pictures of the full chambers in your neighbours property, as evidence for the cause of your issue, and I would be asking to speak to, (or get the contact name of), someone senior at the Water Company when asking they reattend. Tell them you have engaged a Contractor, (on the advice of their employee who initially attended), who has informed you it is not Private and in fact is a blockage in the shared sewer, in a neighbouring property.
Pay the Contractor, as he has attended in good faith, but get an invoice stating that the shared section was blocked, and they have advised you it is a shared sewer so therefore the Water Company's issue. Then go back to the Water Company, present this invoice and ask they settle your contractor's invoice as their employee had previously stated it wasn't their (Water Company's) responsibility.