Dunst's fiance Jesse Plemons is also in the movie starring alongside Benedict Cumberbatch.
The cast and crew agreed to be tested prior to departure from their home base, on arrival in New Zealand and once the 14-day isolation period had been completed.
They were then expected to stay in New Zealand for between two and six weeks depending on their role and workload and had been scheduled to re-start shooting on June 22. Prior to the country's lockdown they had been in the country since January and all held current visas.
Granting an exemption to someone such as a nanny was a risk and there maybe perception risks of whether some of these workers are essential,
a paper said.
"A challenge will be ensuring that the workers are genuinely highly skilled and their roles can't be sourced in New Zealand."
A MBIE spokesperson confirmed that Twyford granted the nanny exemption.
Dunst and her family had spent most of lockdown in New Zealand renting a house
with a lawn for their 2-year-old son Ennis to run around. However, they returned to Hollywood when the lockdown eased.
Meanwhile Cumberbatch is understood to have spent lockdown at Summerlee Luxury Retreat in Te Awanga.
Dunst is not the only one to be allowed to bring her nanny into the country. INEOS Team UK was also granted exemption to bring in a nanny.
It will bring 86 workers and 128 family members as part of the 36th America's Cup.
Meanwhile UK family Barbara Genda and Harry Jarman have been refused an exemption on humanitarian grounds by the Ministry of Health (MOH) to enter New Zealand's maritime border and earlier told the Herald they believed New Zealand's border was a door opened by money.
Their 14-year-old son Eddie was killed on August 9, after he was hit by a speedboat on Moorea Island, near Tahiti, while checking an anchor.
The family wanted to enter New Zealand to sell their yacht so they could return home to the UK before the cyclone season arrived, but are now stranded in French Polynesia.
Hollywood star Kirsten Dunst spotted filming Jane Campion movie in Oamaru
Covid-19 coronavirus: Kirsten Dunst and Jesse Plemons denied special managed isolation request