The short answer is that there is no timezone at the poles. This means that time on the North Pole can be anything that you want it to be. It is anytime and no time at the same time.
Allow me to explain
A timezone is just a convention
See timezones are agreed upon and assigned in a way that is convenient to the people of each country (some countries have more than one timezone.). Timezones are assigned in a way that corresponds to the position of the Sun in the sky. When the sun rises in a particular place, the time there is set to 5 AM for example.
We can assign any timezone to any place on earth but it would be inconvenient. Imagine if the time at the midnight was 8 AM. That is not practical
Timezone is a concept to help people organize their lives and coordinate with others. People can agree to use any timezone they wanted.
The sun sets once a year at the North Pole
Because the Sun sets and rises only once a year at the poles, "time of the day" becomes meaningless. It really doesn't matter.
"Time of Day" has no Meaning at the North Pole
So how do people set up meetings and appointments at the poles?
By default and convention, people at the poles choose GMT as their timezone in order to facilitate communications but they can choose any timezone they wanted. For example, scientific expeditions at the South Pole choose to follow their home country's time. You could have two settlements next to each other on the South Pole each having its own timezone.
When people want to meet at the poles, they often indicate the time as well as the timezone. So instead of "dinner at 8 AM at my igloo", it is "dinner at 8 AM GMT at my igloo."
"Time is just an operational ritual, intended to create the illusion of regularity." Katie Weeman