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It would appear as if Fury was telling the truth to the Avengers and that research into the Tesseract was underway since SHIELD got it back at the end of Captain Marvel. However, Phase 2 itself had not started until after the events in Thor, this is backed up by Coulson who repeats what Fury said to the Avengers.

Loki moves to open the chute beneath the cell, dropping Thor to, what for any regular person, would be certain death. Coulson suddenly appears, wielding a large prototype Phase Two weapon.

AGENT COULSON: Move away please.

Loki steps back from the panel. Coulson gestures to the gun.

AGENT COULSON: Do you like this? We started working on the prototype after you sent the Destroyer. Even I don’t know what it does. Do you wanna find out?

_{The Avengers}

The fact that the Phase 2 project is only in the prototype stage would back up the theory that Phase 2 itself was relatively new but that research into the Tesseract has been going on for a while.

Note the beginning of the conversation when they uncover the plans for Phase 2 and find that SHIELD has been gathering a lot of material around the Tessearct. This is not exactly an overnight process so must have been going on for a while.

TONY: And you’ll get your cube back, no muss, no fuss. What is Phase Two?

Steve enters, and slams a large HYDRA gun on the table.

STEVE: Phase Two is SHIELD used the Cube to make weapons. (to Tony) Sorry, computer was moving a little slow for me.

FURY: Rogers, we gathered everything related to the Tesseract. This does not mean that we’re-

TONY: I’m sorry, Nick. Tony turns the screen towards Fury. It shows a plan for some type of missile. What were you lying?

_{The Avengers}

Fury then goes on to say it was the appearance of the superheroes, supervillains and people like Thor on Earth that made them want to start looking into weapons technology. It wasn't necessarily aliens on their own which made them want to look into weapons technology.

BANNER: I’d like to know why SHIELD is using the Tesseract to build weapons of mass destruction.

FURY: (pointing at Thor) Because of him.

THOR: Me?

FURY: Last year Earth had a visitor from another planet who had a grudge match that leveled a small town. We learned that not only are we not alone, but we are hopelessly hilariously, out-gunned.

THOR: My people want nothing but peace with your planet.

FURY: But you’re not the only people out there, are you? And, you’re not the only threat. The world’s filling up with people who can’t be matched, they can’t be controlled.

STEVE: Like you controlled the cube!

THOR: Your work with the Tesseract is what drew Loki to it, and his allies. It is the signal to all the realms that the earth is ready for A HIGHER FORM OF WAR.

STEVE: A higher form?!

FURY: You forced our hand! We had to come up with some-

TONY: Nuclear deterrent! ’Cause that always calms everything right down.

_{The Avengers}

That said SHIELD has had the Tesseact since the events of Captain Marvel and have probably been researching into it since then. But even with all that time they aren't actually ready to properly develop stuff out of it. It's possible since the events of Captain Marvel that they have simply been trying to control the Tesseract and haven't actually properly managed to do that yet. We see in The Avengers that they still don't have any control over it.

FURY: We prepared for this, Doctor. Harnessing energy from space.

SELVIG: We don’t have the harness. My calculations are far from complete. She’s throwing off interference, radiation. Nothing harmful, low levels of Gamma radiation.

_{The Avengers}

COUNCILMAN: Which is why you should be focusing on Phase Two, it was designed for exactly this purpose.

FURY: Phase Two isn’t ready; our enemy is. We need a response team.

_{The Avengers}

While this isn’t a comment from Marvel, I think the films themselves avoid any discrepancy.

1. Maybe they didn't wait

SHIELD doesn't start producing weapons from the Tesseract until after the events in Thor

We don't know for certain when SHIELD started working on the weapons we see in The Avengers. They could have been working on them since Goose barfed up the Tesseract.

It might have taken them that long just to get them working. Aside from the difficulty of harnessing the power of an Infinity Stone, as we know from Captain America: The Winter Soldier

Arnim Zola, who created Tesseract-powered weapons in World War 2 for the Red Skull, was working within SHIELD in some capacity, but as a HYDRA operative. He could have been deliberately holding the research back.

 2. And if they did, here's why

If we assume Fury actually was telling the truth about the weapons timeline (which is credible! See TheLethalCarrot's answer and vote it up!), it's not a discrepancy. Here's what he says:

FURY: Last year earth had a visitor from another planet who had a grudge match that levelled a small town. We learned that not only are we not alone, but we are hopelessly, hilariously, outgunned.

THOR: My people want nothing but peace with your planet.

NICK FURY: But you're not the only people out there, are you? And, you're not the only threat. The world's filling up with people who can't be matched, they can't be controlled.

The Avengers - IMSDb

So the recent emergence of powered individuals like Iron Man and the Hulk on Earth was motivation in itself, regardless of extra-terrestrial threats.

On the space-y side of things, the Destroyer's attack in Thor caused a lot of very public damage, after which Thor himself disappeared. SHIELD had no way of knowing whether the Asgardian threat was actually resolved (due to the Bifrost being destroyed during Thor's victory over Loki) — for all they knew, an Asgardian army commanded by Loki could show up at any moment.

That's more of a prompt to develop weapons than the events of Captain Marvel, where Fury knew from working with Carol that the Skrulls were in fact

just trying to get their people (and probably also the light speed drive) off earth, and far away to find a new home;

whilst the Kree were only interested in hunting down the Skrulls, and considered C-53 (Earth) something of a backwater s—hole of no interest.

No significant damage was done to Earth, or witnessed by any civilians, so there was no public awareness to create pressure to develop a response, unlike the New Mexico incident.

Given the solid intelligence that neither the Kree nor the Skrulls had any further interest in earth, SHIELD could reasonably have prioritised energy research over weapons until the Avengers-to-be started causing their early-21st-century ruckuses.

This is the timeline for the Tesseract and its use by humans:

c. 1942: Red Skull recovers the Tesseract from Norway. Red Skull's HYDRA faction begins developing weapons using it as a power source.

c. 1943-45: the main conflict between Captain America and HYDRA. The Tesseract is lost in 1945 with Captain America. Howard Stark recovers the Tesseract.

What happens between 1945 and the late 1980s with the Tesseract is currently unclear. Howard Stark, worried about the dangers it might represent, might have kept it locked away within SHIELD until Lawson/Mar-vell took it to power the Light Speed Engine, whether it was given to her or she secretly stole it and Project Pegasus didn't know it was the power source is unknown. Or he might have secreted it away keeping it from everyone and gave it to her. Or she might have stolen it. Who knows at this point. Whatever the case, as of 1989 it wasn't available for human use any more until sometime after 1995 when Goose harked it up on Fury's desk.

Now, again this is hypothetical, but given the secrecy involved, Fury might not have been aware that the item he now had was the same one that powered HYDRA weapons in the 1940s. Stark was killed in 1991, and Peggy Carter, as well as everyone else who'd been around and knew firsthand the Tesseract powered weapons and might have stayed with SHIELD, should have been retired or dead by 1995. So, as far as he knew, the only thing the device did was provide power for a propulsion system that might or might not be useful for weapons. And, in any event, what happened in 1995 wouldn't have necessarily driven the need for radical weapons development: the Skrulls and Kree had left, and based on Fury's first-hand knowledge, they didn't need to develop any special weapons to deal with them on an individual level: he'd seen a Skrull killed by purely mundane means (the one pretending to be Coulson) and had no particular reason to believe the Kree would be any harder to deal with. Further, the Skrulls had left and the Kree only came to Earth because of them, and based on Carol's knowledge, they didn't give a crap about Earth otherwise.

(Incidentally, Coulson never saw a Kree aside from Vers, so he had no reason to know they were blue; or it might have been erased from his memory as a result of Project TAHITI, which is why he didn't recognize the Kree body years later on Agents of SHIELD.)

So the only threat that Fury knew normal portable weapons couldn't deal with was Carol Danvers herself, and it's quite clear for the admiration he has for her by the end of the film he doesn't consider her a threat.

So, he's got a power source that he might not know could be used for weapons (because he doesn't know what it is), and nor particular reason to develop more advanced weapons.

Then 2011 in New Mexico happens (Thor) and he realizes how hilariously outgunned humanity really is. Now there's someone normal weapons can't deal with.

Again, hypothetical: he goes scrambling to see might work, and looks at those weapons SHIELD inherited from the SSR that were captured from HYDRA, and finally makes the connection between Goose's little gift and the power source from the 1940s, and pulls Selwig in (Thor credit stinger) to start making advanced weapons again.

It seems apparent that SHIELD didn't have the capacity to harness the power from the Tesseract until Fury meets Dr. Selvig and Dr. Foster regarding their discoveries in Thor:

Nick Fury: [greeting him] Dr. Selvig.

Erik Selvig: So, you're the man behind all this? It's quite a
labyrinth. I was thinking that you had taken me down here to kill me.

Nick Fury: I've been hearing about the New Mexico situation. Your
work has impressed a lot of people who are much smarter than I am.

Erik Selvig: I had a lot to work with: the Foster theory, a gateway
to another dimension... it's unprecedented...

[Fury doesn't respond]

Erik Selvig: ...isn't it?

Nick Fury: Legend tells us one thing; history, another. But, every
now and then, we find something that belongs to both.

[Fury opens a suitcase, which contains a small cube]

Erik Selvig: What is it?

Nick Fury: Power, Doctor. If we figure out how to tap it, maybe
unlimited power.

[Loki is seen in a reflection, smiling]

Loki: [whispers to Selvig] Well, I guess that's worth a look.

Erik Selvig: Well, I guess that's worth a look.

Thor (2011)

BOLD ITALICS Emphasis Added by me