Does the time difference between Auckland and London change during the year?

Yes. Because of daylight saving time the gap moves between +11h and +13h.

When is the best time to call London from Auckland?

There is no overlap between standard working days, so one side has to call early or late.

Auckland is 11h ahead of London. This gap is not fixed: daylight saving time moves it between +11h and +13h during the year.

Auckland

New Zealand

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London

United Kingdom

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A 9-to-5 working day in Auckland does not overlap one in London at all — somebody has to take the call outside office hours.

City	Time zone	UTC offset
Auckland · New Zealand	Pacific/Auckland	UTC+12
London · United Kingdom	Europe/London	UTC+1

What is the time difference between Auckland and London?: Auckland is 11h ahead of London. This gap is not fixed: daylight saving time moves it between +11h and +13h during the year.
Does the time difference between Auckland and London change during the year?: Yes. Because of daylight saving time the gap moves between +11h and +13h.
When is the best time to call London from Auckland?: There is no overlap between standard working days, so one side has to call early or late.