Telecommers:

BT landline - second line to household, two phones branched off it.

After intermittent occasions of continuous ringing, previously cured by picking up and hanging up, the continuous ringing is now ..continuous.. and incurable by hangup.

Worse, BOTH phones on the line continuously ring.
(The primary line is unaffected).

Worse yet, BT claim there is no fault on the line.

Obv. it is not the fault of the phone as both would not develop the fault together. Thus the fault has to be on the line or exchange? or on-property BT junction box?

Any advice before I have to call them out and risk getting fleeced for £129 whatever (I'm a professor and don't make anywhere near that per hour.. fair pricing??).
{Thanks in advance :}

Sorry no expert as I was power maintenance and construction so didn't get to do the sexy telecomms stuff. However if you consider that problem from an electrical perspective, it sounds like you could be right when you say it's "on property". I suspect you must have checked the wiring in your extension boxes (LJU1 or LJU2's I suspect).

As stated I'm no expert but only use the basic pair (blue/white & white/blue) connected to 2 and 5 in the NTE5. Always done this and never had any problems, although I have noticed provider contractors and builders often connecting orange/white (bell), solid brown and solid blue as they see fit.
The website below may have your answer:

http://www.telephonesuk.co.uk/wiring_info.htm

Looks like, according to them, terminals 2 and 5 have been swapped between two of your extension sockets. Can't say this is the answer but hey, worth a try.

Merry Christmas