If signallers have to 'talk a driver past a signal', does the signal control centre tend to look like the older air traffic centres by using paper train codes laid out on the desk and move the paper train along to the next signaller?
Not quite that bad, but the signaller has to cancel out the train description from the TD berth at the red signal, and then put ("interpose") that description in the TD berth at the next signal. If the red signal is that signaller's last signal, then they need the next signaller's permission before authorising the train to pass it, and then the first signaller tells the second signaller what's coming, and the second signaller interposes the description.
Even if these two signallers are sitting right next to each other in one of the larger signalling centres, they're encouraged to use the telephone for that conversation - so that it is recorded on the Voice Recorders.