On the ex-GNR south of Peterborough, going by the telegraph circuit code cards provided for the instruments/circuits in many of the area's boxes, the only boxes that I know of which appear to have not been telegraph-equipped were Hertford North (opened approx. 1923/4), and the nearby Block Huts of Bayford, Stapleford and Watton
(article about Block Huts can be found in the main site, here : http://www.signalbox.org/branches/sg/index.htm ) : Virtually all other boxes were telegraph-connected at one time or another
(although I cannot say if this was so of the branch lines from such as Huntingdon and Holme).
Perhaps the above Hertford line exceptions were not telegraph-equipped as, for this post-1922 LNER opening of the completed double-tracking, and raising to passenger standard, of this (perhaps regarded as secondary) GN/LNE line from Cuffley to Langley Jn. it had been decided that new lines would rely on telephone communication.
However there were also one or two Goods lines-only boxes in the London area for which AFAIK, no definitive telegraph code has so far been established [e.g. Ferme Park South Down (near Harringay) and East Goods Yard (Finsbury Park) ], so they were perhaps not telegraph-equipped, though it may seem difficult to believe that they were never on the telegraph network.
But from the same circuit cards, virtually all boxes did have telegraph call-codes, so in the late 1960s either were, or had previously been, on at least one telegraph circuit : Some had instruments for, or had a telegraph circuit concentrator and a single sending instrument
(see photo/text in the article, as linked-to by Russell Maiden, available here : - http://www.signalbox.org/branches/jh/telegraph.htm ), on several circuits ; I recall that in at least one location there were five.
There was great variation in the distances over which telegraph circuits stretched according to intended use : E.g. KC-BN (Kings Cross Teleg. Office to New Barnet North) had most boxes on it and seems intended for comparatively local use, while KC-FO reached to Peterborough (actually Eastfield yard Office?), connected relatively few places along the way and was used for John's previously-mentioned Up train reporting to Kings Cross box.
There were several circuits along some stretches of the line south of P'boro' (I don't know about north thereof), not only because of differing types of use, but also as there would have been too much telegraph traffic for just one circuit in the days when the system was also used for stationmasters' telegrams, traffic working messages (e.g. 'Stop traffic to X'), Yard wagon reports of vehicles present by their individual numbers, and relief staffs' duties for 'next week'.
Quite a few of the more important places were on several circuits.
As to the query over separate Up and Down circuits; yes, owing to the presence of some boxes signalling lines in only one direction (sometimes coined 'one side' boxes), there were some circuits which, though many boxes on them signalled all lines in their location, were obviously designed to cater for messages re trains in only one direction: I'd say that the most obvious examples were the YF-HT [Wood Green No.3
(Down box) to Hitchin Teleg. Office], and AU-HT [Wood Green No.4
(Up box) ] to Hitchin Teleg. Office].
Some boxes' signalmen never needed to use their instrument(s) though may have read (or ignored) messages being passed between other boxes. Other boxes may still have had instruments but they were no longer working - this could be that the particular instrument or its line was not working, or that the circuit itself was partially or completely 'down'
[e.g. at either Hatfield No.3 or Welwyn Garden City (I forget which, or both), the telegraph concentrator still had a needle for the XO - DU circuit (Hatfield Teleg.Office - Dunstable), though the line beyond Blackbridge (Ayot) had closed some four years earlier]Towards the late-1960s, at least some signalman vacancies were still advertised (internally) with one of several code letters to indicate certain knowledge or ability(ies) required, as defined in the vacancy list footnotes.
One of these (I think it may have been 'E'), stated, IIRC, "Telegraphist required", and vacancies in the queried line's boxes normally included this, but as the decade came to an end, there was an increasingly serious shortage of signalmen in/near the London area
(there were still about 42 boxes involved in working the main line's traffic in the approx. 32 1/2 miles to Hitchin out of Kings Cross; approx. 23 of these were in London in the first 5-and-a-bit miles : The 'Hertford Loop's 23-ish miles had another 10 block posts).
So to help fill the gap, quite a few signalmen were enticed in from lower graded boxes elsewhere in the Eastern Region, and rather than have too many boxes left switched out to adequately cope with the timetable's busier times, non-telegraphist men were sometimes appointed to boxes customarily required to perform telegraph 'receiving' or 'sending' regularly, presumably with no telegraph training on offer; the resulting loss of some 'wiring-on' by telegraph of trains probably being judged as the lesser evil.
Some circuits saw little, or virtually no use, while others were used up to several times per hour, for regular sending of passing times of certain classes of train to stipulated boxes ahead (e.g. as has been mentioned, to Kings Cross (BX) from Crescent Junction (Peterborough : CJ), Hitchin South (HF) and Hatfield No.1 (HD) ], while much other train reporting was by exception; i.e. not done unless a train(s) was late or running in the wrong order.
For some unknown reason, I particularly remember that one evening at Wood Green No.1, a Down express departure from KX (1S70, 22:15 I think) was sufficiently late departing from there that a Class 3 Parcels booked to follow was run first from Holloway, passing Wood Green while the express still had not got under way.
So it was necessary to advise several further afield boxes of this, and so I had to send, (I think to Potters Bar, Hatfield No.1, Welwyn Garden City and Hitchin Yard) : -
"one s seven o No (pause) three s one five 'xx' " -
- (1 S 7 0 No = train 1S70 not yet offered on the Block ; 3 S 1 5 'xx' = the time, in mins. past the hour, when 3S15 had passed).
As resignalling loomed on the horizon and progressed to staged implementation
(the Oakleigh Park and New Barnet area was first in 1970, and in 1971 I think, Hertford North, and commissioning of the first stage of Peterborough power box), it was preceded by the separate traditional 'Control Office phone' system communicating with the boxes and important stations being steadily replaced by a more modern dialled system (PBXs I think) which also provided dialled 'phone communication between all the 'outstations'.
And so with that, and further spread of non-trained staff, plus other circumstances already mentioned, the instances of telegraph use understandably decreased, non-repair of faults accelerated, and the system finally effectively faded away, almost completely unrecorded, into oblivion.
[ Edited to add further information, clarity, and removal of outdated reference to GNRS :
16:27 14/01/15, and 14:45 15/01/15, and 23:48 20/01/17.]