The date the consuls took office varied: from 222 BC to 153 BC they took office 15 March, and from 153 BC onwards it was on 1 January.
source listed as E.J. Bickerman, Chronology of the Ancient World (Ithaca: Cornell University Press, 1968), p. 64
The day of the election which was made known by an edict, three nundines beforehand (Liv. III.35, Liv. IV.6, Liv. XLII.28), naturally depended upon the day on which the magistrates entered upon their office. The latter, however, was not the same at all times, but was often changed. In general it was observed as a rule, that the magistrates should enter upon their office on the kalendae or idus, unless particular circumstances rendered it impossible; but the months themselves varied at different times, and there are no less than eight or nine months in which the consuls are known to have entered upon their functions, and in many of these cases we know the reasons for which the change was made.
This source then proceeds to list the various dates when consuls began their terms, and the years when those dates were changed.
From B.C. 509 to 493 on the Ides of September.
From B.C. 493 to 479 on the Kalends of September.
From B.C. 479 to 451 on the Kalends of Sextilis.
From B.C. 451 to 449 on the Ides of May.
From B.C. 449 to 443 or 400 Ides of December.
From B.C. 400 to probably till 397 Kalends of October.
From B.C. 397 to 329 (perhaps 327) Kalends of Quintilis.
From B.C. 327 to 223 unknown.
From B.C. 223 to 153 Ides of March.
From B.C. 153 till the end, the Kalends of January.
But modern historians tend to be skeptical about the history of the early Roman Republic, so I don't know how accurate the list of dates is.