So, you already have flow & return pipes from the fire to the cylinder upstairs. That's good, you're halfway there.
The simple answer is yes, you should certainly be able to connect the system in such a way that your "excess" heat from the fire would supplement the boiler output rather than being Drained off as it is now.
Well worth getting a boiler guy take a look.
You have both heat sources being taken to the cylinder at the moment (boiler plus fire.) I guess you have something like a Y-plan system with a motorised valve that directs hot water to either the cylinder or the radiators. If I'm right, that's fine, but it doesn't let the hot flow from the fire into the rads. Thus the fire does not contribute to the heating.
Should all be just a case of re-arranging the system in the airing cupboard. Well worth doing.