Well, the Metropolitan Museum gives us a definitive answer of "it depends", mostly because in practice armour was not done by a single armourer in a single stretch. If, however, we turn to modern reconstrution for an estimate, we get get something of an idea on how long would just the manufacture take.
The time of manufacture will vary wildly. The related question on swords illustrates this well; If you want your one-of-a-kind, mystical, magical katana blade, it will take weeks or months for everyone to meditate before work, tweak every little detail and then get things done just right. If youre interested in a battle-ready weapon of war that will do just as well in a fight, come back in three days.
The way of making these also changed significantly during the middle ages; in the early medieval period, there weren't all that many people who would need armour, and a blacksmith would make maybe a couple of sets over his lifetime. In the late middle ages, some places would have specialists who would churn out "munitions-grade" armour, that could be issued to troops or just stored away in an arsenal for future needs.
By definition, it's impossible to give a general answer to the question, because it will depend on what exactly type of armour we are talking about, whether the specific armourer has made it before or not (so he either already has the necessary tools and experience, or needs to experiment some), whether he has the necessary materials at hand and how busy he is right now.
What I can give you, though, are estimates based on modern reconstructions, which tell us the minimum time to make a set of munitions-grade armour, from the moment it is ordered to the moment it's handed over to the customer.
This largely depends on three things:
- What technology you're using
- What materials do you have available
- How ornate it should be
I'm going to ignore the ornamentation part, because you can drag that out pretty much arbitrarily.
I'll further be assuming that the armourer already has a fully equipped workshop available, along with a couple of helpers (most craftwork, especially smithing, relied on semi-qualified help being available to work the bellows, help with heavy hammerring, etc.), and that they had made the exact type of armour before, so there is no need for prototyping or experimentation.
With that out of the way, going by technology:
In Europe, these would usually be made out of cowhide. This was a pretty universally used material, so it would have been perfectly possible to just buy it. The leather can be cured in several ways to improve durability, which also factors into the production time: you can have plain (or painted) leather, cuir bouilli, or laquered leather (the old Japanese way).
There is a number of designs for leather armour, but for a rough idea, the opposite ends of the spectrum are leather "plate" (in other words, a cuirass) and leather scale, one of the most complex of the latter being Japanese kozane.
The nice thing about leather is that you have the cow make it for you. This in turn means that manufacture time of individual pieces is largely independent of their size, and depends rather on their complexity. So, to make a piece - be it a cuirass, pauldrons, greaves, bracers or whatever, is going to take roughly the same amount of time.
Call it four hours for cutting and shaping, another four (with the help of apprentices) to sew/rivet it together and once you have all the pieces done, four hours for assembly and finishing. If you want it painted, boiled or laquered, add about another day between the shaping and the sewing; the process itself does not take that long, but sometimes has to be repeated and you need to wait for the pieces to dry. In total, call it three days to a week.
This construction of armour was pretty popular for a multitude of reasons; if you don't have extremely tough leather, layering it in this way improves its resistance to piercing immensely, and you can make it out of essentially scrap leather.
The representative type for this would be a kozane (small-scale) based do maru, which used ingenious construction to get the most out of crappy material (they were initially made out of leather with some metal scales, and eventually transitioned to all-metal as it became available).
The production pipeline for this goes like this: first, you need to get your scales. You need about a thousand of those, but you can get a cookie-cutter and a hole punch and churn them out pretty quickly; you could have the requisite amount within a work day.
After that, you lay them out in rows, tie them using leather lace (like this), and give them about three coats of laquer. This will take a day or two; the rows will stay in the shape you put them in, so you need to be already shaping them, and again you're waiting for the laquer to dry.
Next, you make some fittings (the non-kozane parts, essentially) and get to lacing it all together. Expect to use up a couple hundred meters of lace and about five days to get everything done right (kebiki odoshi is a lot like knitting).
This gives you a total of a week or more for just the cuirass (do - also includes the dangly kusazuri) and the pauldrons (sode). The other parts need to be made separately using whatever other technique you prefer.
This one depends heavily on the size. Today you can buy maille links, back then you would have to make them yourself, so you need to factor that in; of course, you could have a bunch of them pre-made if you were so inclined.
Besides size, construction times also depend on what type of links you're using. In butted maille, the rings are just bent and closed in a circle and you hope for the best - this was used in Japan and by cheap Europeans, and is the fastest to make. Riveted maille is best in terms of protection, but more difficult to produce. Welded or punched links would be used together with riveted ones to speed up production.
For a hauberk, you'd need something in the neighbourhood of 15000 10mm links (more or less standart size, as much as this can be said in an era with no standarts whatsoever). Assuming you have wire (not unreasonable), you can make that many butted links easily in a day by winding the wire around a rod and then cutting it with a chisel.
For riveted links, you first make the butted ones, than flatten the ends, and then you punch rivet holes. A manufactory of six people could concievably produce about 500 of those per hour (along with the requisite rivets). This gives you about 30 work hours (call it three days) to make the requisite amount.
The most practical way to produce a hauberk is to have a bunch of maille squares pre-made and just link them together (with the requisite expansions etc.), but if you have to make everything from scratch, re-enactors give a figure of about three man-weeks for a butted hauberk. This can be parallelized if you have a whole workshop going at it, but you're probably not going to manage it under a week, unless you're working from pre-fabricates.
For riveted maille, the work will be roughly three to four times slower, if you have two-person teams working on each square, and they don't make too many mistakes; call it three weeks best-case, otherwise something more like twelve for a single armourer and an apprentice.
This one again depends more on complexity than on size. You need to draw plates from ingots at the begining, but that doesn't take that much longer than cutting them our of leather. The reason why breastplates were so prolific is that with the proper metallurgy, they're pretty quick to make.
An equipped manufactory could churn out one like this in about two days; multiply that by the number of parts to get a rough idea for simpler sets. For the really complex ones (although note that most of the ones that look like this are renaissance rather than gothic), the production really can take months, because everything has to be individually tweaked and fitted to make sure you can actually move.
Leather: 3-7 days for a cuirass with accessories, upwards of a week for scale cuirasses without accessories
Maille: For a fully manned manufactory, a week plus for a butted hauberk, upwards of three for a riveted one
Plate: Two or three days for a breastplate, plus another two or three days for each accessory. Months for a full movie-style knight set, if you want to be able to move.