Firstly Jesus was born before 4 BC (when Heriod died) using the current dating system so the celebrations are for his nominal birth date at the end of 1 BC and the start 1 AD. Note that there is no year zero - BC and AD are measured using ordinals (first, second etc) and not cardinals (one, two etc.). So 2000AD represents the two thousandth year which will not be completed until the end of the year.
The date of the birth of Christ was calculated by a monk called Dionysius Exiguus in 525 AD. He assumed that the Annunciation (when the angel Gabriel appeared to Mary) was on the spring equinox in the year 752 AUC (ab urbe condita or the foundation of Rome) and used this as the basis of his dating system (anno ab Incarnatione). The spring equinox was on 25th March with the year beginning at the start of that month (hence the names of September through to December or 7th to 10th months). The birth would be 9 months later on midwinter day 25th December (there were only two seasons, Winter and Summer, which started and finished on the equinox and hence midsummer's day is June 24th). Thus the birth date was 25th December 1 BC - the start of 1AD (modern) was 8 days later - day of the naming and circumcision. (Using January 1st as the start of the year is still called stilo circumcisionis).
The English calendar was established by the Venerable Bede at the start of the eighth century. He used the Dionysius year count, but with the autumn equinox starting the year.
This system was used until the 10th century when the start of the year was changed back to St Mary's feast day (25th March). However Bede had added 6 months in moving the start of the year to September, and another 6 months were added in moving it back to March. The original Dionysius dates are measured from 25 March 1 BC (known as the Pisean style) while the English dates are measured from 25 March 1 AD (Florentine style) - one year after the Annunciation. In 1752 the English New Year was moved back to January 1st (thus 1751 had only 282 days) which had been used throughout the rest of Europe for two centuries (Scotland changed in 1601 so Elizabeth 1st died in 1602 in England and 1603 in Scotland). Prior to 1752, if a couple married in late March, April or May their first child could be born in January, February or early March of the same year.
The Gregorian reform of the calendar was implemented from 1st January 1582 in most of western Europe which became the start of the new year, but not in the British Isles. Pope Gregory XIII corrected the Julian year of 365 days and 6 hours by 11 minutes and 14 seconds by skipping the leap year if the year was divisible by 100 unless it was divisible by 400. England changed to the Gregorian calendar in 1752 deleting the 11 days from September between the 2nd and 14th (1752 had 355 days). Thus Christmas should have occured 11 days later (January 6th also known as Old Christmas Day), but the date of December 25th was retained. The date of the equinox and solstice now occur on the 21st or 22nd (depending on the leap year cycle), but midsummer's day remains on the 24th June. The start of the financial year which had coincided with the calendar year now starts 11 days later on April 5th. Note that the different countries in Europe have moved to the Gregorian calendar at different times. The Eastern Orthodox churches still use the Julian Calendar where every fourth year is a leap year and so are now 13 days behind the Gregorian calendar. The oriental Orthodox churches use a modified Gregorian calendar where century years are only leap years if dividing by 9 leaves a remainder of 2 or 6.
So to conclude the nominal date of Christ's birth is 25th December 2 BC, and hence his circumcision or naming would occur on the 1st January 1 BC, so 1 AD was the first anniversary, and 2000 AD was the 2000th anniversary after all.