It all belongs to the
BT Group and its care & attention is within
Openreach's domain.
The grey box is a BT66 and I see two interesting features:
- There is one black cable from the pole that enters the BT66. There are two black cables that exit the BT66. You have already speculated that the second cable could be your neighbours service feed.
- Somewhere within your property there must be yet another junction. Why? There is a black cable that leaves the BT66 and enters your property, yet all the cables seen so far have white sheaths.
When you order a FTTC service from your chosen supplier, you could ask for an engineering visit for wiring normalisation. Ideally, you really need to have the cable replaced from the BT66 to the NTE5/A, in one contiguous length. Some of the existing internal wiring is of dubious quality and, if I remember correctly from when we looked at it some years ago, there is a distinct possibility of split legs in the pair. (Instead of one pair of wires twisted together, there is one wire from one pair and the second wire from another pair.)
All in all, it might be best to find a local ex-GPO/BT/Openreach person via the classified advertisements and have her/him perform the re-wire (BT66 to NTE5/A).