Kings Lynn is about 40 miles from Peterborough, I live 9 miles north of town.
You can get a bus from Peterborough which is direct, the train is slow and goes via either Ely or Cambridge.
Flying would really only be practical into cambridge or Norwich, Stansted is a good distance away and no direct connections to Kings Lynn.
Train to Peterborough would be ideal, no changes required from
Edinburgh or
Glasgow. Its a long and tedious journey on the A1, especially north of Alnwick where the dual carriageway ends for long sections which can be painfully slow. I'd go via Jedburgh across country from
Newcastle if driving, lovely long and quieter roads and more scenic too.