Just out of curiosity. I remember reading that Kalashnikov did just that with the help of German and Russian designers. (Of course he had Stalin to contend with as well.)
Is it possible or impossible?

Money and resources make most things possible including meeting your timeline. It would take a company with considerable capital who was willing to assume the risk.

No. it would take the legal dept 4x that time frame (easy) to hammer out the CYA portion of production.

I'd say easily yes with the right minds, attitude, and funding.  During Operation DESERT STORM, the GBU-28 laser guided bomb design was prototyped, tested, deployed and used.  They went from the ”go” decision for the specific bomb design on 13 Feb 1991 to the first bomb dropped on target on 27 Feb 1991.

"Technical improvisation in times of war is an artform, and a very good metric of where real capabilities lie. Those who can cobble together a vitally needed resource at short notice, with limited time and budgets, are worthy of admiration - their efforts more than often produce decisive results."  Dr. Carlo Kopp

They designed the SR71 or the U2 in like a year or so.  With the right resources and money it can be done.

china yes America probably not.

question is would anybody want it.   most likely full of mim shit.

Yes, magpul did this with the Masada in time for shot show.
recent article in American Rifleman about it.  IIRC took about 10 hours to decide on features in a single meeting.

But did it come to full production? From concept to tool room prototype to full production is a different animal.
I've worked in firearms manufacturing for 20 years total. It takes a hell of a lot to bring a rifle to full production.  Machine fixtures to be made, and usually modified at least once.

Anything is possible. If your whole team is on the same page and you have a giant pile of money then the likelihood increases exponentially.

According to the book "The Gun" the process was much longer than three months.  It involved the usual testing of multiple designs. Tweaking the designs after the trials.  And then more trials. More tweaking.  And more trials.  Until they arrived at the AK-47.  Then they had production issues that made them use a milled receiver instead of sheet steel receiver of the original design.  Sorry but the AK didn't happen in 3 months.

Define "production".  

It may have taken a long time for the ACR fo come to market, but I imagine MagPull could have pulled off a slow- or limited-production rifle if they wanted to.

Sure.  All a question of money, effort, and inspiration.  I can't imagine the Sten gun took much longer.  Guide Lamp Division of GM designed and built a million Liberty pistols is three months (granted a very primitive device).  North American Aviation had the P-51 flying in about that amount of time.

The NA-73 prototype's airframe was completed 153 days after the contract was signed ( 10 April 1940 to 9 September 1940)  but the aircraft itself didn't fly for another 47 days ( 26 October 1940), due to delays in engine installation, etc.   That's 200 days total from contract to first flight - still impressive though for a very complex aircraft.  

However it then crash landed on 20 November 1940 and did not fly again until 11 January.1941.  The first production Mustang Mk Is were not delivered until February 1942 - 22 months after the contract was signed -  and did not enter combat until May 1942 - 25 months after the contract was signed.    Given that the original goal of the British purchasing commission was to have North American produce the Curtis P-40 under license, and start cranking out P-40s in about 3 months, it was not an unqualified success from a contract management perspective.




When Apple approached US companies and asked them to produce curved edge glass screens for it's iPhone s it was told it would take at least 9 months as the manufacturing technology was apparently both new and a bit tricky.   Apple wanted them in 6 weeks, so they checked with Chinese companies and they were told "no problem".  

The Chinese make this happen by trucking in what ever number of workers are required, putting them up in dorms and working long shifts to get the work done.  It ends up being competitively priced as the wages are low, and there are no additional expenses such as EPA requirements, etc.

It's not a model that will work in America as quite frankly we don't really want to go work in a dorm for long shifts at very low wages, nor do we (despite all the Republican posturing and bashing of agencies like the EPA) want to live in a country with the massive air and water quality problems and related health issues that China now has.  Life is very cheap in China and it's a great example of how bad totally unrestrained capitalism can be.

The other thing people mis understand about China is the quality of the products.  Chinese firms have the ability to build very high quality stuff.  If they build crap it's because that's what the specifications called for by the company that it selling it.

As an example you can buy scuba diving regulators of the same basic design with various brand names on it, but those different brands won't sell for the same price as they are not all the same in terms of quality.  Quality still costs money and the companies that spec the regulator with tighter tolerances and narrower performance specifications pay a lot more for that level of quality.  The companies who spec the regulator with wider tolerances and wider performance specs can get them made for a lot less money - in part because all the rejected parts that don't make the cut for the higher specification regulators can be used in the lower spec branded regulators.


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The end result is that yes, you could very likely get a new firearm design into full production in 90 days, and you could also have it made to very high quality standards - but it wouldn't be inexpensive,

The old saying still applies:

You can get it cheap,
you can get it fast,
you can get it done right,
but you can only have 2 out of the 3.


 

It was a PBS documentary on Stalin that is where I believe the 3 months were given to come up with a rifle. I could be wrong indeed.

Sure - just look at US made Sig pistols. They're still trying to iron out their quality control issues.

These guys couldn't do it in 3 months, but got close.

The prototype NA-73X was rolled out in September 1940, just 102 days after the order had been placed; it first flew on 26 October 1940, 149 days into the contract, an uncommonly short gestation period even during the war.

I have been in industry as a design engineer, a manufacturing engineer, and now a quality engineer, for a total of 40 years.  I have a bit of knowlege and expereince here related to the OP's question.

The short answer is --- it depends.  Depends on what, you say?

1.  Definition of "volume production".
10 units per day is FAR easier to achieve than 3,000/day (which was the peak production rate for Garands at Springfield Armory in WW2)

2.  Type of construction.  
If the design is complicated (like the Garand, for example), there are literally THOUSANDS of tools, jigs, and fixtures that would need to be made in order to repeatedy build good parts, and lead times on that stuff, under the best of conditions would measured in weeks, if not months, to construct them.

On the other hand, as others mentioned, if you are building STEN guns, tooling is minimal, parts are very simple, and some pieces could be outsourced as off-the-shelf hardware items.  But STEN guns are not rifles, so the design is not a good benchmark for comparison.

Even a design of a rifle geared around using simple tooling (the AR-18 is a really good example of this) would require forming dies and welding fixtures for the sheet metal pieces, and I don't think any "class A" tooling for that kind of stuff could be done in less than about 6 weeks, and that would be at a premium tool construction price rate.

So if your design is simple, and you are willing to pay premiums to get tools fast, 12 weeks to START production is feasible.  It might still take a couple of months after that to get the process stabilized in volume.

On the other hand, if you are making M14's, or G3's, you will need probably six months to get the tools constructed, and another 3-6 months to debug the tools.   Getting production for one of those designs up and running steadily would take at least a year.

Winchester designed and built the first test prototype M1 Carbine in 14 days.
After testing turned up some issues, Winchester made the final production design in 34 days.

In November 1940 Winchester designed the M1 Carbine cartridge to a War Department request.  It was in at least limited production before May 1941, and had been for months because a number of companies submitted proposed carbines for testing in May 1941.  
None of these first submissions were acceptable, so Winchester designed their carbine to be submitted with the next batch.

In August 1941 Winchester began the final Carbine design and submitted it in September 1941.  It was approved for full production on October 23, 1941.
First tooling-up samples delivered by Inland in November 1941.

So, Carbine ammo requested in November 1940 and experimental guns being built using the ammo within a couple of months.
First prototype Winchester Carbine designed and built in 14 days.
Final production design done in 34 days, adopted in September 1941 and approved for full production in October.
First production samples out in November 1941.

In US it takes 3 months +2 weeks...

Inland delivered the samples in November of 1941, but the first month they actually produced more than 20 units was June of 1942, when they delivered 361 units.   They didn't produce over 10,000 units per month until September of that year.   Once they got to over 10K, I would say they had achieved "volume production"