Obtain from the Local Authority Tree Officer a copy of the TPO order. This will give the information of the trees and the owner of the trees. Ask the Tree Officer if they have received any tree Reports in the past.
You then have several options open to you. You can contact the Owners of the trees and raise your concerns. Do this in writing so there is a record of this. You can on your own contact a tree company who understand English Tree Law and By Laws.
Forget about contacting Solicitors, they are clueless when it comes to Tree Law and anything within by laws about trees, trespass of trees, dangerous trees etc.
Please note if you contact your own Tree Surgeon they by law have no right to climb the tree as it belongs to your neighbour, only a ground based assessment can be carried out until permission is granted by your neighbour. Also a Tree officer is NOT ALLOWED to climb or gain access to the tree crowns other than by having his feet on the ground. Simply a Tree Officer is not insured to leave Terra Firma. However a tree Surgeon feels there is sufficent evidence that the Tree's are dangerous then they can climb, and can under certain circumstances make the tree/s safe. But make sure you know the law before you go tree hoping!
You mention the trees have a heavy lean. A good tree company can view this lean from the ground and give you a rating of the potential zone factors related if failure incurs, i.e, if the failure of a tree is in the striking zone of your Mothers home due to failure from a poor specimen then this would equate to a HIGH RISK. This should then be passed on to the owner of the trees, the TO and if needs be your Solicitor.
Just because both parties want to sell up and move does not remove the problem, in fact it can raise a serious issue with both properties. As you mention your Mother wanting to move, then simply ask your Mother's estate agent for a tree report from your neighbour. This will then bring the matter to a very abrupt head where you then have legal rights over your neighbour in obtaining a report. Mortgage lenders are becoming more and more aware of trees and there problems. Any potential buyer of either home would be notified of the TPO'd trees, and a lot of mortgage companies would want a report to make sure there investment is safe.
Use the law to your advantage, but be mindful most of what you need to know comes from the brains of a good Tree Surgeon and not the Tree Officer, and certainly not some high street solicitor who usually make matters a lot worse.
And finally after reading your post the felling zone is in the owners of the trees garden. Give them the problem, and let them worry about it, but get a good Tree Surgeon on board to help you.