TNG "Family" is our best source so far. Jack Crusher's holomessage with the TWOK uniform is dated at Wesley's birth, hence 2348. His uniform came to feature TNG-style collar pips before his 2354 death, though, as Wesley inherits those.
It doesn't get much narrower than that. Adding to the above, Janeway's father wears a uniform that clearly belongs to the TNG family rather than the TWOK one, in a hallucination of dubious accuracy, but that's only in 2358.