Assuming this refers to the character, not the actors, we have dates for a few based upon on-screen evidence:

  • In "The Tomb of the Cybermen", the Doctor estimates his age at 450 years. Assuming not too long has elapsed since his regeneration (given the presence of Jamie McCrimmon, who joined soon after), that suggests the First Doctor changed around his 450th year.
  • In "Robot", the newly-regenerated Fourth Doctor says that he is 748 years old.
  • In "Time and the Rani", the just-regenerated Seventh Doctor says outright that he is exactly 953 years old. This is the only verified age for a Doctor at time of regeneration to date.

At some point between "Time and the Rani" and "The Day of the Doctor" (War Doctor's perspective), the Doctor began counting his age differently. We do not know why, how or when. The TARDIS (per "The Doctor's Wife" appears to have adopted this new count, too.

  • In "The Day of the Doctor", the War Doctor states he is 800 years old. He regenerates during this story, meaning the Ninth Doctor was 800 years old when he started.
  • In "Aliens of London", and other Series 1 stories, the Ninth Doctor states that he is 900 years old. He travels with Rose Tyler for a time before regenerating, and based on comments by the Tenth Doctor it's implied that the Doctor was still about 900 by his new accounting when he changed.
  • In "The End of Time", soon before his regeneration the Tenth Doctor says he is 906.

In spin-off media, promotional material for "Spearhead from Space" referred to the Third Doctor as "500 years old". The "The Brotherhood" (comic story) set right before the Second Doctor's regeneration had the Second Doctor stating he was 500.

The "Cold Fusion" (novel) had the Fifth Doctor state that he was 813 years old when he last regenerated (in "Logopolis").

The "Vampire Science" (novel) had the Eighth Doctor state he was 1009 when he last regenerated (in "The TV Movie").

We therefore do not know the ages (by either old or new accounting) at which the War and Twelfth Doctors were "born".