This is really just putting together others' pieces (so if you are inclined to upvote this, go and do likewise to HollyLeaves's and puzzle-guesser's answers first). I think you are
My prefix is unbroken — yet doubled here —
A hypothetical person, or beginning of years;
What is one is not broken in two. 1 appears twice in this number. If one refers to "one", as in this sentence, one is speaking of a hypothetical person. And years are counted up from 1 AD/CE and down from 1 BC/BCE.
My infix is doubled in all of the most recent times:
When split in thirds, pronoun, plural, and conjunction one will find;
Recent years begin with two thousand. And THOUSAND = THOU+US+AND.
My suffix is of one without friends;
You might hear it when an (informal) timed contest is at an end;
"One is one and all alone and evermore shall be so", as the old song has it. And a contest may end with a countdown ...3...2...1 (though I'd have thought usually it would end at 0 rather than 1?). [OP mentions in comments that there's also a one/won pun here.]
My whole: spoken the second it starts,
Comes sooner, and is less, than the sum of its parts.
Perhaps the idea here is that one + thousand + one = 1002, which of course is bigger than 1001 and comes later if you're listing numbers in the usual way? [OP mentions in comments that this is also alluding to counting "one thousand one, one thousand two, ..." in games like hide-and-seek.]
Finally, there's allegedly a "hidden hint". So far, the best thing I've been able to come up with for that is
the repeated use of "My", suggestive of the Roman numeral MI which of course is 1001.