As of 2021, there have been three New Year Specials. The first time a Doctor Who special aired on New Year's Day was The End of Time: Part Two, which aired a week after Part One, the 2009 Christmas Special.
For series 11, showrunner Chris Chibnall made the decision to cut the yearly Christmas Day tradition, which had been ongoing since 2005, when BBC Wales Doctor Who began, in favour of this special. Resolution, the 2019 New Year Special, was the first to be primarily set on New Year's Day. The End of Time: Part Two previously featured a short scene near the end, set on 1 January 2005.
In Doctor Who spin-offs, the very first episode of The Sarah Jane Adventures (Invasion of the Bane), as well as the Torchwood series 1 finale (Captain Jack Harkness / End of Days), premiered on 1 January 2007. Invasion of the Bane, both produced and broadcast separately from the rest of its season, might be considered the first New Year Special among the Doctor Who "family" of shows.
Though not a television special, the very first episode of Doctor Who to premiere on 1 January was "Volcano", very nominally the eighth part of the 1965-66 serial The Daleks' Master Plan. The first parts of the 1972 serial Day of the Daleks and the 1977 serial The Face of Evil also premiered on 1 January, although, while the episode of The Daleks' Master Plan actually celebrated the occasion, the latter two were just a case of the calendar happening to fall on that date and made no mention of it within the story itself, much like the fourth episode of The Invasion falling by pure happenstance on the series 5th anniversary yet letting the occasion pass without celebration.
Part one of Spyfall, the opening episode of series 12, premiered on New Years Day 2020. Although not promoted as a special, the episode was purposefully aired outside of the series' usual time slot in order to coincide with New Year's Day.