Aang would have a heroic blue screen of death, then be wracked with guilt for the rest of his life. Him ever mastering the Avatar State would be very difficult because of what happened to Hama. He'd never return to the Southern Water Tribe, although they'd probably send out people to search for him (being the world's last hope and all). Kanna would make a raft out of debris and paddle onwards to the South Pole, because the only thing that can bring down Gran-Gran is Kryptonite.

Aang might still eventually master the elements and defeat the Fire Nation but it would take years longer. He'd pull his punches against his enemies, having killed somebody already. There'd also be a lot more heat on him than in canon. Since Roku is still a figure in living memory of a lot of people, Fire Lord Azulon won't leave hunting Aang to his favorite toy general/admiral/princess like Ozai did. Azulon will see Aang as an existential threat to the Fire Nation, and without the training and protection the Southern Water Tribe offered him Aang will be in danger all the time.