Only once before has England scored over 218 in the last innings to win a match against NZ in twelve games, in Christchurch, 96/97 season. In England, the highest successful 4th innings by an English squad is that 218, back in 1965 at Lords. They scored 211 to win in 1999 at Birmingham - in fact, they have only done such a feat six times in England.
96/1 - Birmingham, 1965
218/3 - Lords, 1965
138/2 - The Oval, 1969
138/3 - The Oval, 1978
118/3 - Lords, 1978
211/3 - Birmingham, 1999
At the other end of the scale, of course, NZ have only won four matches in England and only in one of them was England batting last, where England scored 162 all out at The Oval in 1999.
My money would be on England breaking their record and making the 282 runs, unless (and I would love for it to happen) New Zealand can pick up the same kind of swing that England did. Which I have no idea if they're capable of doing because I'm still learning about swing as we spe- er, type.