The Platform 5 books dated 1986 list 19 Mk.3b coaches (11083-11101) as having a 'train telephone', and just two HST coaches, 41041 & 41066, which were formed in the Yorkshire Pullman and Tees Tyne Pullman sets. I do remember us coming back from London once, and it must have been on one of the Pullman HST rakes as a family friend went down to the telephone to try the phone out by calling his wife on it!
11083-11094 were the ones branded 'Pullman', whilst 11095-11101 were in effect general FOs (as were non-telephone fitted 11064-11072).
41041 and 41066 were joined by approximately half the WR TFs - 41003-41037 (odd No.s), 41121-41145 (odd No.s) and 41164.
The FOs and TFs had the phone fitted in place of one luggage rack, and could be identified by the fact there was no metal bar splitting the frosted window like those in toilet compartments or behind luggage racks.
A number of Mark 4 FOs were also fitted with payphones. My 1994 Platform 5 book shows 11201, 11203, 11204, 11207, 11211, 11213, 11214, 11217, 11219, 11221, 11222, 11225, 11227-11229, 11231, 11233, 11235, 11237, 11239, 11242, 11243, 11245-11247, 11249-11263 (odd No.s) as being fitted.
Mark 3 TRBs (402xx series), TRSBs (404xx series), TRFBs (407xx series) and the unique TRFM (40619) all gained payphones adjacent to the buffet.
Mark 3A RFMs (102xx) and Mark 4 RFMs (103xx) were also fitted with payphones adjacent to the buffet.
Mark 2F RFBs (12xx series) had payphones fitted in First Class, so Standard Class passengers had to walk along he side corridor, through a swing door and the phone was straight ahead.
Mark 2F RLOs (67xx series) had payphones fitted, but unlike the RFBs this was in place of the luggage rack that wasn't required in a Sleeper lounge car.
Mark 1 RBRs (1647, 1671, 1683, 1686, 1689, 1691, 1692, 1696, 1697, 1699) used on InterCity Anglia services had payphones fitted, as did three Mark 1 RMBs (1842, 1850, 1871) used on CrossCountry services.
If I recall correctly, as others have said, the phone looked like a standard BT payphone that took BT phonecards and they ran off the Cellnet network (Cellnet being BT's mobile network, that became O2 before being spun off). If I also recall correctly the individual coach number was the last four/five digits of the phone number of the payphone. I usually used them to make a quick call 15 minutes before my train was due in to my home station to ensure I would get picked up for a lift home. Minimal call due to the cost.