Northern Norway is much nearer than you think; it is a 90-minute flight from Oslo to Bodø, and London-Tromsø takes three and a half hours. The most northerly airport in the world, Longyearbyen, is less than three hours from Oslo. You can fly direct to Northern Norway from London, Stockholm, Helsinki and Murmansk, and there are hundreds of departures every week from Oslo and Southern Norway to the bigger airports in Northern Norway.
There is a train from Oslo via Trondheim to Mo i Rana, Fauske and Bodø, and from Stockholm via Boden and Kiruna to Narvik. In the far north, there are buses from cities in Northern Sweden, Northern Finland and Russia to Northern Norway. The most elegant way to arrive in Northern Norway is to sail with a Hurtigruten ship from Bergen; it takes three days to sail to Bodø, four to Tromsø and five and a half to Kirkenes.