Given the state of the national telephone system in England at the time, would the signals companies of the TF maintained its own, independent communication system, or relied on the exisitng national system as it was at the time?
Like their counterparts in the pre-war Expeditionary Force, the basic formations of the Territorial Force, whether infantry divisions or mounted brigades, were independent of the sort of army corps that played such a big role in the tactics, logistics and administration of the French and German armies. Nonetheless, in cases where two or more divisions were given a mission that required close cooperation between (or among) them, they were formed into a uniquely British formation known as the 'army (group of two or more divisions)'. As the 'army' had no responsibilities related to logistics or administration, the units of 'army troops' assigned directly to its headquarters were, for the most part, signals units that helped the general officer commanding the 'army' to keep in touch with the divisions and mounted brigades assigned to him.
Each grouping of three Territorial Force signals companies seems to have been intended to provide for the communications needs of one of the five Territorial Force 'armies' that were to be formed in the event of war. Thus, for example, the three companies stationed in Liverpool would provide a cable company, an air line company and a wireless company to the Western Army.
As far as I can tell, these armies were designed to have a somewhat regional character, with initial zones of responsibility that corresponded (at least roughly) with the five 'commands' of the Regular Forces. (The sixth 'command' of the Regular Forces was in Ireland). In the event of a small landing in its zone, the 'army' in place would take charge of operations, making use of its own resources, as well as divisions and/or mounted brigades provided by other 'armies'. If, however, the small landing became a large one, and, as a result, a war of grand manoeuvres began on the island of Great Britain, then some of the 'armies' might become as mobile as the 'armies' of the Expeditionary Force.
Because of the last-named scenario, and the possibility that a hostile army might destroy key parts of the civilian telephone and telegraph system, the signals infrastructure of armies had to be both independent of the civilian system and reasonably mobile. Thus, while I am sure that the forces mobilised for home defence made extensive use of civilian communications systems (after all, you can never get enough bandwitdth ...), and there were points of connection between the Territorial Force system and the civilian system, there was much more to the 'army troops' signals companies than posting men in telephone exchanges and telegraph offices.