the first question would be whether the land definitely does belong to the neighbouring landowner or not. It can often be difficult to determine the exact position of a boundary in terms of centimetres or inches but a plan should enable identification of land in terms of several feet usually. As above, properties are generally registered with a plan for identification purposes showing the general position of a boundary. Sometimes it can be relatively straightforward to determine the location of the boundary by reference to another structure such as the house but not always and even then not necessarily with the degree of accuracy that may be required.
From what you say, of you has been informed that the land does not form part of the title and you seem to accept this position though I make the above statement simply to observe that it is not necessarily determinative based upon the title plan itself and if there is doubt in this respect, then a boundary dispute process may be worth considering.
If there is no question that some or all of the land in question belongs to the neighbouring landowner, then you need to revisit the information that was given to you by your solicitor originally. As above, it is not the responsibility of a solicitor to check the boundaries for pragmatic reasons that is not cost-effective for solicitors to visit properties. The solicitor can discharge their responsibilities in respect of boundaries and by ensuring that they sent you a copy of the file plan and asked you to confirm that the plans were in accordance with the physical boundaries as you inspected the property. If they did this, and you felt to identify any discrepancies, I fear it is unlikely you would have a claim against the solicitor in this respect. Conversely, if they did not send you a copy of the plan then there may be negligence in this respect.
As above, if part of the land does indeed belong to a neighbouring landowner, then you either need to consider a boundary dispute process or if you are able to contact the previous owners who live there for at least seven years or more, then you may be able to contemplate a claim for adverse possession if they are willing to cooperate in this respect with you. Otherwise, you may need to consider negotiating with the adjoining landowner as regards ***** ***** of the land which of course may involve the payment of money