You might have done this trip already, but if you haven't then I'll recommend to do the Copenhagen - Helsingør - Hillerød as planned. From Hillerød to Roskilde, I would try to stay on smaller roads, probably something like through Lynge, Ganløse, Søsum, Veksø, Gundsølille and Store Valby. For the last bit, from Roskilde to Copenhagen, I would take the train.
If you do the trip, then you can decide in Hillerød if you want to continue to Roskilde, or just take the train back to Copenhagen. Train goes every 20 minutes. You can check train schedules here: journeyplanner.dk
Btw. you can bring regular bikes in most danish trains. Not the metro, though. And it's free in all s-trains in the Copenhagen area.