If yr neighbours statements directly contradict their neighbours statements and this goes to land Tribunal, you can ask for them to be cross examined on their statements.
Check your house insurance because these issues are often covered by legal expenses insurance.
You don’t have a claim against the solicitor unless the solicitor failed to carry out an enquiry which is unlikely to be the case.
However you might have claim against the sellers if they didn’t declare that the drive was shared on the Sellers Property Info Form. It is a specific question. It would be worth contacting them to see if they ever agreed to share the drive . Their statement is the best that you can get.
Limitation on contract claims last 6 years from date of contract knowledge if later so you might be out of time. After 6 years the solicitors might have destroyed their file.
It is worth getting letter from the sellers.
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