in answer to your question ,that's exactly what is happening.
The hot gases from the fire are diverted (via the chimney flue valve) [ the
eyelet sticking down at the back of the fire] under the back boiler causing
the water in it to heat ,this water ,unlike a modern system ,is directly fed
into the cylinder upstairs .
So if the fire is raging and the temperature rises above 100c then ,as we
all no (allowing for latent heat) water boils.
(Happy memories of my Nan yelling 'Go run off the hot water before the
bedroom fills with steam')