Which months were added to the Roman calendar last?

January and February. The oldest Roman calendar had ten months beginning in March, and simply ignored the roughly sixty days of winter as a period when nothing agricultural or military happened - an unnamed gap rather than months.

The fossil evidence is in the names we still use. September, October, November and December mean seventh, eighth, ninth and tenth, and they sit in the ninth to twelfth positions - two out, exactly as you would expect if two months were later inserted at the front. July and August were renamed for Julius Caesar and Augustus, having previously been Quintilis and Sextilis, the fifth and sixth.

New Year's Day moved to January at various points and not everywhere at once. England kept 25 March as the start of the legal year until 1752, which is why old documents carry double dates for January to March, and why the British tax year still ends on 5 April - it is 25 March plus the eleven days dropped when Britain finally adopted the Gregorian calendar.