Have you checked out who has what rights?
You may have a copy of your title from when you bought the property or details in your report on title from your solicitor.
If not you can obtain copies of the register and plans for both your property and the neighbouring one for a small fee from here...
http://www.landregisteronline.gov.uk/. Your neighbour's postcode can be found at www.royalmail.com.
Who owns the shared access? One of you with rights for the other over it or do you own part each? It's difficult to say without knowing the laout/rights/ownership etc...
The register will also give you the name and address of the landlord who you could write to about the noise and disturbance etc...asking him if he could ask his tenants to be more considerate. Depending on the tenants you could first try talking to them or writing them a note (keep a copy) asking them to be more considerate. If this doesn't work the you could copy this to the Landlord with a letter to them.
If there any way you can stop the gate from banging so much?