The Air Monks never decide to send Aang to the Eastern Air Temple, and as a consequence Aang never gets iceberg'd and dies in the Fire Nation's extermination of the Air Nomads. Though the Avatar is immediately reincarnated into the Water Tribe, the Airbenders are now irrevocably gone, and the cycle has been broken: within another century or two the Avatar will reincarnate into the Fire Nation, and after that life cease to exist.
The unnamed Water Tribe Avatar (perhaps Pakku?) goes on to fight in the war against the Fire Nation until he, too, eventually falls in combat. The Avatar is then reborn into the Earth Kingdom...
As a certain little blind girl.