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.

You can suck as much water out *** u can with a wet dry shop vac. If that doesn't let air flow u will have to go under and put a small hole in the lowest point to drain it. Then coer it up with duct tape. U might have to replace that section of pipe.

Both answers are valid. If possible remove the vent and use the shop vac. You can buy some hose and tape it to the end of the shop vac to extend it's reach... it will need to be air tight so seal it well. A duct cleaning service may be able to help as well, but try running the furnace fan and use the shop vac first.