Because of the Amundsen-Scott base at the South Pole, and how it is supplied, local time at the South Pole is New Zealand time. The North Pole has no local time since there's no human presence there.
Air traffic generally uses UTC for all purposes, and if there's no clear local time, it's highly likely that UTC will be chosen. Wikipedia's page on Air France 447 (which crashed in the Atlantic Ocean on route from Brazil to France) exclusively uses UTC for its crash timeline notes, while its page on MH17 uses local (Ukraine) of its shootdown and for MH370 uses Malaysian Time for its timeline.
Given that Alert is the northernmost continuous human settlement, I'd imagine that any search-and-rescue effort for an extreme polar crash would be coordinated out of there. Alert's timezone is (US/Canada) Eastern Time.
However, this is a trick question. To get an extreme polar flight, you're looking at something like LA-Dubai. No one on that flight would be bringing enough winter clothing to survive a crash at the North Pole, and so everyone would be dead of frostbite or hypothermia if not killed by the impact itself, and thus there are no survivors.
It was not long ago that a 747 crossing the pole had all four engines die at the same moment. After losing tens of thousands of feet of altitude, they got the engines started again.
Rolls Royce (or was it GE?) figured out what exact combination of intake speed, pressure, humidity, temperature, and history caused the failure, and prevented it on subsequent flights. It had been considered (!), but they thought it would never be encountered on any real flight. The engines restarted because, at lower altitude, conditions were different enough.
The pilots found the event distressing. It is possible most passengers didn't notice.
If you’re not familiar with it, I was riffing on a popular riddle: “if a plane crashes precisely on the border of two countries, where do you bury the survivors?”
