First off thank you for the most informative and comprehensive replies. I'll try to answer your questions.
A bit of background. When we moved in the previous owners had two phone lines, one terminated where the ADSLNAtion is now, and the other line in the study, where Infinity is now installed. After we moved in we wanted just one line, BT came to remove some ISDN socket (or something, I can't remember what exactly), that was on the second line in the studu. The upshot was we ended up with one phone line and two NTE5 sockets that were both live, so were connected in parallel across the line.
Fast forward to ADSL. It was self install ADSL, the modem was put in the second NTE5 location in the study. The line was a long distance from the exchange and picked up a lot of RF interference at night, so I fitted the ADSLNation face plate to the first NTE5, and fed the second NTE5 with the unfiltered output, and at the same time fed the one RJ11 phone socket, in the lounge from the filtered output. It did away with dangly filters and the blurb said it was a better filter for longer lines.
Fast forward to Infinity, the engineer didn't look at the ADSLNation faceplate, he was shown the study where the ADSL modem was was, saw it was plugged into a 'BT logo' NTE5 and without looking anywhere else fitted it as you can see in the pic.
And here we are.
The ADSLNation is still live, it feeds the one phone socket in the lounge as well as the study, I've added pic3 to show its location better, it would be so inconvenient to have the Infinity socket there, and I don't think Mrs Edward would allow a dangly wire from it, to round the corner somehow into the lounge.
I think the best solution is to remove first NTE5, crimp the incoming to the cable that goes up to feed the Infinity socket where it is in the study, use a spare pair in that cable (3 pairs in it), to return the filtered output back down to the hallway and crimp to the wires that feed the phone socket in the lounge. How does that sound? All can be done in the recessed socket where ADSLNation is now, then put a blank cover on it.
You asked for a pic behind the ADSLNation, I can't just now, it would cut the phone and Internet off, but I can assure you it is just as described above. I've also added a pic of the NTE5 removed from the study to make way for the Infinity faceplate.
It's an amazing offer to pop round with the barrow, sadly I'm in Leicestershire, or pints would have been on me. I'm so reluctant to bring this to the attention of BT, and would much rather be in control of what happens...if you are passing by this way any time soon, I'll gladly take you and whoever is with you out for lunch, we are in the heart of the country some very nice places to visit
Hope that all makes sense.
PS Edited to add some line stats.