In all the Indian Defenses, in general, Black allows White to occupy the centre with his pawns.

Nonetheless, in Nimzo-Indian and Queen's Indian Defense Black has many moves to attack White's pawn centre very early in the game, thus White cannot gain too much space.

King's Indian Defense (and Gruenfeld Defense), on the other side, really do not care that much for the first 4-5 moves of White, so that he can occupy the centre with an "horde" of pawns (see for example the [Classical Variation][1], the [Saemisch Variation][2] or the [Four Pawns Attack][3]), with a huge advantage of space.

Since [hypermodern principles][4] in chess were born only in early 20th century, they were not universally accepted before '40s. And having such a big space disadvantage in the opening was seen with suspicion (if not explicitly considered "strategically losing"). That's why, in my opinion, KID had to wait to be considered consistent enough to be played regularly by top players.

  [1]: http://en.wikipedia.org/wiki/King%27s_Indian_Defence
  [2]: http://en.wikipedia.org/wiki/King%27s_Indian_Defence,_S%C3%A4misch_Variation
  [3]: http://en.wikipedia.org/wiki/King%27s_Indian_Defence,_Four_Pawns_Attack "Four Pawns Attack"
  [4]: http://en.wikipedia.org/wiki/Hypermodernism_%28chess%29