Thanks for your enquiry.
It appears to me that your old Solicitor has clearly been negligent and has breached his duty of care towards you, as any Conveyancing Solicitor would have known to include a right of way for all parties using your party (unless of course it was expressly agreed that it should only benefit you, which does not sound to be the case).
In order to make a claim you must be able to show that a loss has been suffered and that the loss was caused by a breach of duty of care.
You must be in time to bring a claim. Time limits ( called the limitation period) have been laid down in statute for different types of claims. In most professional negligence cases, the time limit is six years from the date of the negligence but sometimes, when the negligence has not become evident until later, this time limit may be extended. In your case, as you have only become aware of the issue, you will be covered under this rule.
You should therefore contact another litigation Solicitor to pursue a claim. A lot of Solicitors will act on a "no win no fee" basis and you should perhaps find a Solicitor who is willing to do this. Your Solicitor will advise you more fully, but in these circumstances, it would normally be resolved by your neighbour entering into a further Deed (if they are willing to do so) so that it includes a right for any visitors to use the right of way, and your old Solicitor would be responsible for paying the required amount which your neighbour asks for to sign this Deed, provided of course, it is a reasonable sum.
All Solicitors have to have professional indemnity insurance, and therefore your old Solicitor will therefore place the matter in the hands of his Insurers.
I hope this assists and sets out the general legal position.
Of course, if your neighbour is willing to sign a further Deed, to expand on the rights granted, all the better!