England captain Alastair Cook said the ongoing uncertainty over the identity of the team's next permanent coach would not bother him so long as his side kept winning.

As reports emerged on Monday that Australia's Trevor Bayliss had emerged as the front-runner to replace the sacked Peter Moores as England coach rather than compatriot Jason Gillespie, the long-time favourite, Cook's men were completing a stunning 124-run win in the first Test against New Zealand at Lord's.

"Obviously it would be great if we do have a new coach," said Cook after England went 1-0 up in the two-match series.

"We're a little bit in limbo but if we're a little bit in limbo and keep playing like that I'll be happy."

This match was a triumph for Cook who, having ended his two-year wait for a Test century against the West Indies in Barbados earlier this month, made 162 -- his first Test ton on home soil since 2013 -- in England's second innings at Lord's.

He was also able to watch from the other end as Ben Stokes struck the fastest-ever Test hundred at Lord's, off just 85 balls, on a sensational Sunday afternoon.

England, 134 runs behind on first innings, eventually set the Black Caps a target of 345 on Monday's final day before dismissing the tourists for 220.

Runs under pressure

Stokes, born in New Zealand but brought up in England from the age of 12, starred with the ball too by taking the key wickets of Kane Williamson and Black Caps skipper Brendon McCullum in successive balls on Monday during a return of three for 38.

"I've never heard Lord's sounding like that, as it did when Ben got those two wickets," said Cook.

But Cook said the 23-year-old Durham all-rounder's first-innings 92, made after England had collapsed to 30 for four, and Joe Root's scores of 98 and 84 had also played their part in keeping England in the game, along with his own nine-hour century.

"When the team is 130 behind you need someone to go and get a really big score -- luckily it was me in this game," said the left-handed opener.

"But the partnership between Ben and Joe in the first innings, to play as well as that in those conditions under that kind of pressure, I think got the lads believing a little bit."

New Zealand, with Trent Boult taking nine wickets in the match, were on top for large parts of the game and their first innings 523 featured Williamson's composed 132.

Yet still they were well-beaten.

"To England's credit they were too good for us in the crucial stages but I'm still really proud of our guys and the way we kept giving," said McCullum.

And the skipper insisted there would be no change in New Zealand's bold approach come the second Test at Headingley starting on Friday.

"It (defeat) hurts a lot, I won't lie," he said.

"At the same time there's an element of pride that we continue to play a style of cricket that gives us our greatest chance (of winning)."

"We will highlight that we have been playing well, we scored over 720 runs and took 20 wickets (at Lord's).

"We ticked off a lot of things that would normally lead to a Test win but some of the key moments we weren't quite able to come out on top," McCullum added.