The poles are on every merdian though. As the poles are infinitely small anybody standing on it would overlap onto other time zones. they would have to decide which time zone they overlap most and use that one.
To be pedantic, they almost certainly use UTC at the North Pole, since that's the international scientific standard anyway. However, unless fractions of a second are important, UTC and GMT can be considered to be the same anyway:
http://en.wikipedia.o...inated_Universal_Time
I would suggest that time does not cease to exist at the poles or any other infinitely small point. (let's not get started about Black Holes and relativity and all that as it is not relevant here).
In relation to the specific question, the meridians are only used as a method of measurement of time on the earth. They are used to relate our clocks to the rotation of the earth in a convenient way because we “tell the time” by the position of the sun in the sky as the earth rotates.
However, at the poles, our clock time is still the same as at the equator and to which point on the equator (or anywhere else) you relate your clock is largely irrelevant. There are still (roughly) 24 hours in a day and still (roughly) 365 days in a year. It’s just that the sun does not rise and set in the “traditional” manner when you travel beyond the Arctic or Antarctic Circles. But time still passes just the same.
the further you are from the equator the more diurnal variation you get. You go from 12 hours of light to 6 months of light. So whether you'd actually need to know what 'time' it was if you lived at the North Pole is hard to say. You'd probably just go by your body clock and eat/sleep when you felt like it.
OK, imagine this scenario. A research station at the South Pole is expecting a delivery of supplies. The plane leaves Tierro del Feugo at 0900hrs local time. The flight is 5 hours. At what time in the Antarctic will the plane arrive.