Hi
You say the house is a 60's build IIR the backboilers in these houses was
(as you say) behind a coal fire with a flue diverter in the chimney to suck
heat around it when the fire was lit.
This type of system was a direct system with the same water in the hot
cylinder being routed through the back boiler .(memories of dad yelling Turn
on the Bl*&^y hot water quick before the bedroom fills with steam).
When central heating was fitted one way to heat the existing tank was to
remove the immersion heater and replace it with a heat exchanger coil which
the new gas boiler heated.
Another was to replace the existing cylinder with an indirect unit and
disconnect the back boiler totally leaving the boiler behind the fire empty.
It may be worth checking your cylinder to verify the system.

HTH
CJ.