Account for 1537-8
https://www.british-history.ac.uk/london-record-soc/vol31/pp174-194
To John Houghe, organmaker, for mending the organs this year, 12d
To the said John Houghe, organmaker, for mending a pair of bellows, 5s.
And another one!!
1553
https://www.british-history.ac.uk/early-eng-text-soc/vol128/pp394-396
payed to Ihon howe for mendynge the great organs & mendynge the bell'owes and for mendynge the lyttell' organs, as dothe appeare by a byll'
And more:
1559
https://www.british-history.ac.uk/no-series/churchwardens-st-martin-fields/1525-1603/pp179-195
It'm payd the laste of July to the Orgen maker for tow new skyns (fn. 20) for the bellowes xiijd, for a pounde of glewe iiijd, for fyre jd, for xiiij new springes for the bases ijs iiijd, for latten for the tonges and tow new stoppes ijs, for shew makers (fn. 21) ends ijd, for ther worke for him and his felowe for tow dayes
In Midsummer Night's Dream, typically dated to 1595/1596, a character is introduced as Francis Flute, the bellows-mender.
What is a bellows mender? This is not a new question. Here is an edition of Shakespeare's plays from 1790:
Here is the current Wikipedia entry, connecting the "flute" in his name with the pipe organs:
"Flute's name, like that of the other mechanicals, is metonymical and derives from his craft: "Flute" references a church organ, an instrument prominently featuring the bellows a bellows-mender might be called upon to repair."
Wikipedia also tells us this about pipe organs around 1600:
"In England, many pipe organs were taken out of churches during the English Reformation of the 16th century and the Commonwealth period. Often these were relocated to private homes."
With pipe organs removed from churches, this could be a joke about how someone in that profession wouldn't have much work, and so would need to become an actor. That's a real possibility.
Searching the OED gives some more context. "Bellows" in the sense of instruments used to blow air into a fire, has a very long history in English. Much more recent and much, much less common in 1600 was "bellows" used in the sense of pumping air into a pipe organ.
A search on EEBO produces very very few examples of "bellows" literally related to musical instruments. One comical passage from Nicholas Breton's The Forte of Fancie from 1570, though, does refer to an organ with burst bellows, but it seems to also be a metaphorical reference to the lungs:
Organes, with the bellowes burst, and battred many wayes: his fife, three holes in one: his Harpe, with nere a stringe: great pittie trust mee for to see, so broken euery thinge:
This might even have been an inspiration for whoever wrote the Shakespeare play, since Breton's works are often compared with Shakespeare's and many similarities have been found.
Robert Greene in his 1592 work A Quip for an Upstart Courtier actually uses the word in a very similar sense to Shakespeare. This is the same Robert Greene who died in 1592 but whose Greene's Groats lampoons Shakespeare:
so then he began to tell mee, that by his art he was a skinner, the second said hee was a ioyner, the thirde was a sadler, the fourth a waterman, the fifte was a cutler, the sixt was a bellows mender, the seuenth a plaisterer, and the eight a printeBut this was written before Midsummer Night's Dream, so may have been the source for this character.
There's another possibility. As I mention here, giant bellows were necessary to run blast furnaces, one of which Henry Neville owned at Mayfield in Sussex to produce iron ordinance.
And I've actually found a reference to such an occupation in a book written a bit later Irelands naturall history, 1657, and even mentions the necessity of repairing them.
Here is a description of the bellows. Note the reference to pipes:
And here is the reference to bellows maker:
"a list of whose names and offices here followeth: wood-cutters, who fell the timber; sawyers, to saw the timber; carpenters, smiths, masons, and bellow-makers, to erect the iron-works, with all the appurtenances thereof, and to repair them from time to time"
Interestingly, John Florio in his 1598 World of Words references an Italian word for bellows maker:
manticciaro, a bellowes maker
mantice, mantici, mantico, a paire of bellowes: also the guts whereby euery creature drawes breath