Why was Captain America called The First Avenger, when Hulk and Iron Man were enlisted by Nick Fury before him?
Or are the times between the Hulk & Iron Man movies meant to have taken place after Captain America's return?
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This is a case of marketing hyperbole and careful parsing of the facts. Captain America precedes the existence, as a character of both Iron Man and Thor, in fact, he precedes the existence of all of the Avengers being one of the first and most successful of the Avenger's properties. Yes, technically the Avengers formed and THEN they found Cap but in our real world, Captain America appeared first in print.
Captain America is a fictional character, a superhero who appears in comic books published by Marvel Comics. The character first appeared in Captain America Comics #1 (cover-dated March 1941), from Marvel Comics' 1940s predecessor, Timely Comics, and was created by Joe Simon and Jack Kirby.
Thor first appeared in Journey into Mystery #83 (Aug. 1962) and was created by editor-plotter Stan Lee, scripter Larry Lieber, and penciller Jack Kirby. The mythological Thor had appeared previously in Venus #12-13 (Feb-Apr 1951).
Iron Man is a fictional character, a superhero who appears in comic books published by Marvel Comics. The character was created by writer-editor Stan Lee, developed by scripter Larry Lieber, and designed by artists Don Heck and Jack Kirby. He made his first appearance in Tales of Suspense #39 (March 1963).
The Avengers is a team of superheroes, appearing in comic books published by Marvel Comics. The team made its debut in The Avengers #1 (Sept. 1963), and was created by writer-editor Stan Lee and artist/co-plotter Jack Kirby, following the trend of super-hero teams after the success of DC Comics' Justice League of America.
One major reason why the first Captain America film was subtitled The First Avenger was for the sake of international marketing.
Marvel were concerned that a film called Captain America would not do particularly well in a some countries and added the First Avenger subtitle so that local distributors would have the option to drop Captain America from the title entirely. In the end this only happened in Russia, South Korea and Ukraine. 
There's not necessarily any in-universe logic to why he would be regarded as The First Avenger except for maybe his involvement with the Strategic Science Reserve setting in motion the creation of the Avengers Initiative. However, I suspect that it was simply the best alternate title for the film that Marvel could come up with.
In the Marvel Movie universe canon Steve Rogers was recruited into the Scientific Strategic Reserve, a division of the US military with elements that would be absorbed into S.H.I.E.L.D. As he was technically "avenging" acts against humanity during WWII well before the creation of the Avengers Initiative he would have been considered the First Avenger. Nick Fury used the concept of a gifted individual or group of individuals protecting humanity from threats that conventional means could not achieve. Also if you will remember neither Tony Stark or Thor had agreed to working with S.H.I.E.L.D. until Loki threatened Earth.
Captain America does predate the Avengers, but he was not one of the first Avengers. He was a member of the Invaders with Namor, the original Human Torch and some others. Wasp came up with the name Avengers at the end of Avengers #1 and Cap wasn't unfroze until Avengers #4. All of the original, "first" Avengers had apperances that predate Avengers #1 but they weren't called Avengers until the end of issue #1.