"How do I persuade BT it needs attention?"
This is where life gets interesting, can I assume that BT is both the Communications Provider and the Broadband provider in this instance? Even if this is the case they are seperate entities as far as getting Broadband or Telephony problems sorted. Either way carries the possibility of charges being raised, particularly if you try and get a Broadband fault created.
I would contact your CP and tell them that you are concerned about the condition of the line plant serving you and your neighbour. Tell them that you have a DP with the cover missing exposing the wires. What should happen is that a fault report code of 'DRO' (Damage report other) should be generated and in due course an Openreach technician should show up. With any luck it won't be the one who ignored the problem last time round.
Do not under any circumstances mention the word damage (sorry David, bad idea).
The days of BT/Openreach doing everything for free, no questions asked, are long gone. Any whiff of deliberate or accidental damage anywhere within an End User's curtledge will result in Openreach billing the CP. Almost without exception the CP will pass this cost on to their customer, usually at a considerable markup on what OR has charged them. This is time related and bills of several hundred quid are not unusual if the job takes some time. If the job requires two men, such as renewing an overhead cable that tree surgeons or scaffolders have taken out then the hourly rate is doubled.
If you do go through the above process and the festering BT66 is not replaced, then raise a complaint with your CP.
Feel free to PM me if you need any other info.