FWIW, there are no flights from NGS to NRT, so as Kieron has noted, you'll need to fly into HND and do the ground transfer to NRT. It's a shame none of the LCCs fly to Tokyo from NGS.

I guess another option would be to take the train to Hakata Station in Fukuoka and then the Fukuoka City Subway to FUK but depending on your level of comfort and how much luggage you have, this might not be feasible. From FUK, JL, NH and Jetstar and Skymark all fly to NRT.

It also doesn't seem like there's a direct bus service from Nagasaki to FUK, which would no doubt make getting to FUK much easier.