My own area of study is the armour of high and late medieval Europe. So my answer will focus on that, not on the Early Middle Ages. I mention this caveat because the economics and social organization of Europe were very different between 600 and 1450, and this effected things like how armour was made, which in turn effected the time it took to make it.

My source for this is Alan Williams' The Knight and the Blast Furnace.

A mail shirt might have between 28,000 and 50,000 links, depending on the size of the links and the length of the skirt an sleeves. Some mail was made of alternating riveted and solid links (IE, something like a washer). This was quicker to make, and modern estimates suggest it would take around 750 man-hours to manufacture. If a single laborer worked 10 hour days, this would take 75 days to make (not including sundays and feast days). However, laborers often didn't work alone, and workshops would include division of labor to speed up the process. So the actual time to manufacture a shirt would often be less than 75 days, even if it represented 750 hours of labor - how many people worked on a shirt, and how well they collaborated would determine the actual time of manufacture.

From the 14th century onwards, mail is increasingly made of all rivetted links, perhaps because it allows a tighter weave with thicker links and thus makes mail more protective. Rivetting all those extra links would add around 250 man-hours of labor, for a total of 1000 man-hours.

This made mail rather expensive, as you can imagine. In the beginning of the 14th century mail shirts bought in Bruges in Flanders were the equivalent of 60-130 days wages of a common soldier on campaign. In the early 15th century mail shirts bought from the Westphalia region of Germany were the equivalent of around 25 days wages, which is a good deal more affordable. At least some of this reduction in price may have been due to the re-use of mail - mail is easy to recycle, alter, cut up and repurpose. Many surviving mail shirts shows signs of alteration from decades or more after they were first made, and smaller pieces of mail armour like standards (collars), sleaves, skirts and gussets (underarm guards) may well have been made from older mail shirts that were cut up. So a lord buying mail shirts for his retinue might not be buying new mail, but 'remanufactured' mail.

As a final aside, the first step to making mail is making some form of wire or at least some thin piece of metal that can be bent into a ring. The quickest way to do this is to draw it - basically pulling an iron rod through a series of hole in a 'draw plate', creating a wire of a given thickness. This process is first mentioned by Theophilus in the 11th century, but mail with links of fairly even thickness dates as early as the 8th century. Some medieval mail is made from 'wire' of less even thickness, which may have been made through other processes like cutting strips from flat pieces of metal and then twisting them. I mention the manufacture of wire because while it isn't included in the calculations above, it is important to keep in mind that this was labor that needed to be performed before mail could be made - even though it wasn't necessarily performed in the mailmaker's workshop by the mailmakers themselves. Improvements in making wire made mailmaking faster and mail more affordable.

EDIT: A final note is that mailmaking and making plate armour were different crafts, and at least in larger cities like London were represented by different guilds.