Williamson, Kyle Jamieson and Mitchell Santner, along with the team physiotherapist, were unable to secure an exemption to travel to Britain until May 11 so will remain in New Delhi until then.
The New Zealand contingent involved in the Indian Premier League (IPL) will fly home on Friday, the country's cricket board said on Thursday, but skipper Kane Williamson and three others will stay in India until next week before departing for England.
International cricketers started heading home on Wednesday after Tuesday's abrupt suspension of the lucrative Twenty20 competition, which was triggered by a resurgence of the COVID-19 pandemic in India.
Travel restrictions have prevented the departure of some, however, and organisers have pledged to put on a charter flight to take 37 Australians temporarily to Maldives or Sri Lanka until an Australian government ban on passengers from India is lifted.
There are no restrictions on New Zealanders repatriating and most of the players, support staff and commentators will head to Auckland on Friday, New Zealand Cricket said (NZC).
The players in the squad for a Test tour of England, which includes the inaugural World Test Championship final against India from June 18-22, were scheduled to go straight to Britain after the conclusion of the IPL.
Williamson, Kyle Jamieson and Mitchell Santner, along with the team physiotherapist, were unable to secure an exemption to travel to Britain until May 11 so will remain in New Delhi until then.
Fast bowler Trent Boult, who is also in the Test squad, has been given permission by NZC to join the rest of the cohort on the flight to New Zealand on Friday.
"Trent Boult will return home briefly to see his family before reconnecting with the Test squad in early June, possibly in time to be picked for the second Test against England but certainly in time for the (World Test Championship) final," the NZC statement said.
India remains in the firm grip of the second wave of the new coronavirus with the World Health Organization reporting the country accounted for nearly half the coronavirus cases reported worldwide last week, and a quarter of the deaths.
Eight England players returned to London on Wednesday with four more scheduled to follow them over the next day or so. The players face a period of quarantine in approved hotels.
That fate also faces the Australian cohort after they finally arrive home via Maldives or Sri Lanka, where they will wait out the current government ban on arrivals for anyone who has been to India, which is due to expire on May 15.
Former batsman Michael Hussey will not be joining his compatriots, however, after contracting COVID-19. He must stay in isolation and register two negative Tests for the virus before he is allowed to leave India.
"His symptoms are relatively mild, so he's in for a stint of isolation in his hotel for at least 10 days, but his team have got some really good support systems around him," players' union boss Todd Greenberg told reporters in Sydney on Wednesday.
There were 17 New Zealanders in all involved in the IPL, including 10 players.
