The TV show has now (in the revived series) definitely indicated that the Doctor did have a normally aging childhood. The Doctor had a baby bassinet, played in a nursery, and looked like a typical human eight-year-old human boy when he was taken to stare into the untempered schism at the age of eight. So yes, he did have a normally-aging childhood, just not one that's ever described in notable detail.
Shortly after regenerating, the Second Doctor guessed he was about 450-years-old, but it's hard to say how reliable that guess was, or how much time passed in adventures where there were no human companions to use as a chronometer.
There is also very little firm information available from the TV series (as distinct from novels and other spinoff media) about how Time Lords age, and even the hints are contradictory.
- one aged-appearing Time Lord said he was 'too long in this body'
- the 10th Doctor was artificially aged a century and appeared like a very geriatric, liver-spotted, balding human man.
- shortly after the 10th Doctor had his regenerating cells temporarily neutralized to show that if he could live but not regenerate he'd look like a geriatric garden gnome.
- the Doctor's mental travels may also not quite match his physical travels. When a temporal paradox is removed, he and his companions at the center of the paradox can remember a year that didn't exist.
- the 11th Doctor tells Amy that unlike her, he doesn't age.
- the 11th Doctor's companions were surprised that the 1,103-year-old Doctor at Lake Silencio looked very similar to the 908-year-old version they had seen 6 months prior.
- In terms of the actors, both the 4th Doctor and 10th Doctor visibly aged during their time on the series.
- On the other hand, River Song talked about shaving a few years off her future encounters with Doctor, and her prior appearances seem to prove that she pulled it off.
The information in novels and other spinoff media is even more contradictory -- different writers have had different ideas about this, as they have about other things. The novels etc. Also have contradictory information about whether or not Time Lords have a childhood, which is now a settled question in the revived TV series.
As a result, it's not possible to say "exactly" -- or even approximately -- how he aged enough to appear as he did when we first saw him in The Unearthly Child.
- ↑ which he gave to Amy in "A Good Man Goes to War"
- ↑ which he remembers playing with radioactive blocks in "Smith and Jones"
- ↑ if he looked similar to his friend, the Master, who he remembers undergoing the same ceremony in "The Sound of Drums"
- ↑ This was first confirmed (20 years later) in "The Five Doctors", when he described himself as "the original, so to speak".
- ↑ in Tomb of the Cybermen
- ↑ by the Master in "The Sound of Drums"
- ↑ again done artificially by the Master in "Last of the Time Lords"
- ↑ in the events in "Utopia", "The Sound of Drums" and "Last of the Time Lords"
- ↑ when she tries to seduce him at the end of "Flesh and Stone"
- ↑ and mistakably similar to the 909-year-old they saw in the diner in "The Impossible Astronaut".
- ↑ in "Let's Kill Hitler"