When I try to nap, I'm pretty sure it takes ~5 minutes to doze off usually, but sometimes even more than that.
I think that the key to going right into the nap, nearly as soon as your head hits the pillow, is to have a routine of using a timer, and get up quickly as soon as the time is up. Then, you won't lie there worrying that you'll oversleep or thinking you should be doing something else more important. Your mind is given permission to fall off, knowing it will come right back with the timer beeping.
I've found it helps to cover my head with another pillow, blocking out some light and noise. The head sandwich feeling is also relaxing and may contribute to my falling right out of consciousness in 10 seconds or less.
During meditation retreats, I often fell asleep during the first morning sessions. My teacher said that this was ok, but gave me comparable tip if I wanted to stay awake: meditate standing, when falling asleep, your body will always wake up in moments to stay balanced.
As people already noticed, you have to play a little bit with the timing.