My grandchildren left their bathroom tap running which flooded said floor and nearby hall carpet. Water dripped?/poured? down the ducted heating vent in the bathroom.
What should be done re this? Turn the heater on to dry it out? Or is there electrical wiring which would make this dangerous?

It's going to be nearly impossible to open up the duct and wipe it out. I'd simply turn on the furnace fan and let it blow until you're sure it's dry.