Princess Anne spoke to well-wishers outside the church after attending the service with her father Prince Philip, brother Prince Edward and other royals.
Buckingham Palace said the Queen is "still recuperating" from a heavy cold.
Missing church was a "precautionary measure" in cold and wet weather and there was "no sense of undue concern".
Twelve days ago the palace said both the Queen and the Duke of Edinburgh had heavy colds and had cancelled their plans to travel to Sandringham on 21 December.
They left for their Christmas break a day late, travelling by helicopter instead of train.
Prince Philip has since made a full recovery.
The Queen did not attend  the Christmas Day service at her Norfolk estate, the first time she has missed the service in many years.
The New Year's Day service at Sandringham was led by the Bishop of Norwich.
Princess Anne's husband, Vice Admiral Sir Timothy Laurence, who accompanied her to the service, added that the Queen was feeling "not too bad".
The Duke and Duchess of Cambridge were also absent from the service, after they, Prince George and Princess Charlotte spent Christmas with Kate's family at Bucklebury in Berkshire.
By Royal Correspondent Peter Hunt
For a second time, during the holiday season, senior royals have attended church.
For a second time, the Queen has been a notable absentee.
Given her deep personal faith, it's a decision the Supreme Governor of the Church of England will not have taken lightly.
But at 90 - and on the advice of her doctor - she is exercising caution as she recovers from a heavy cold which she's had for at least 12 days, and which is bound to have left her feeling pretty miserable.
Officials insist the head of state is up and about; she has not been to hospital or had tests; and she continues to deal with the documents she receives in her government red box.
As well as missing church, the under-the-weather monarch has become a voluntary prisoner inside her own Sandringham home.
She hasn't yet had an opportunity to go outside and explore her 20,000 acre private Norfolk estate.
Read more here.
The Queen is feeling "better" despite missing a New Year's Day church service at Sandringham, her daughter Princess Anne has said.