In the Avengers movie, humans tried to kill Loki and the Chitauri by launching a nuclear weapon. However, would this nuclear weapon actually have killed Loki?
Yes, a nuke would certainly kill Loki.
Within the Earth-19999 movie universe, Loki's powers would be insufficient to resist a nuclear blast. Odin's spell may have given him the physical form of an Asgardian (Æsir) but they're only a few times more resilient than the average human. As a Frost-Giant hybrid his tolerance to heat might even be lower than that of the average Asgardian.
Without his forcefield to protect him (and we've seen no evidence that he has this power in the movie universe), Loki's physical strength would be utterly inadequate to protect him from either the heat or percussive effects of a nuclear weapon.
That said, his enhanced physique, super-healing and superhuman immune system would probably protect him from the effects of radiation poisoning if he was outside the immediate blast zone.
Yes, a nuclear weapon would kill Loki.
(This answer includes spoilers from Thor and Thor 2)
In Thor, Odin tells Loki that they are not Gods. He said that they will die just like any other being, only that they live longer than humans/mortals.
In Thor 2: The Dark World, Frigga, Thor and Loki's mother, was killed by a simple stab wound as she was defending Jane.
Also, if Thor, Lady Sif, and Fandral could easily kill the Frost Giants in Thor with weapons like Mjolnir and swords, a nuclear missile would be able to kill Loki.