We don't know for certain, but based on the timing of when blue started to appear as a "regular" writing color (as opposed to a "special" one) we can make a good guess.

We'll be headed to the 19th century, but it might help to look briefly at the process for making iron gall ink (galls are high in tannin -- hides could be used as well, but galls were easier). Pliny the Elder (circa the 1st century) mentioned a process that could have led to the ink, by trying to detect if verdigris (the patina that forms on copper) is genuine: "The fraud may also be detected by using a leaf of papyrus, which has been steeped in an infusion of nut-galls; for it becomes black immediately upon the genuine verdigris being applied." The Codex Sinaticus showed iron gall use in the 4th century (in the southeast Mediterranean), and in the 5th century, Martianus Capella (in Africa) gives a recipe in his Encyclopedia of Seven Free Arts. Iron gall was introduced to central Europe around the 7th century, and as we get deeper into medieval times, the ink reaches common use.

There's a rather good two-minute video here via the British Library explaining its manufacture. I will quibble that the recipe wasn't always exactly the same and their near-black resulting ink isn't always what happened -- it could come down as a much paler brown -- but the important point is that due to oxidization, the ink would turn black either way. Again, due to variety of recipes, you might get an end result of black on the blue or on the violet side, but generally speaking nobody would think of it as a "colored" ink -- there were other recipes for those, the most prominently being used was red, like in this sample from the Iowa State collection.

(Red ink, incidentally, has a level of prominence going back to the Old Kingdom of Egypt, as we have accounting documents that use red and black; not black = positive and red = negative quite yet. Red was numbers that needed to be accounted and remaining balance, and black meant actual receipts/deliveries.)

The process of gall ink involved acidic material. The 1820s also saw steel tips of pens enter wide use, and steel didn't react well to acid. Additionally, there was the problem of faint writing -- again, just like in medieval writing, the color would come down brown and turn black (and eventually, over time, fade browner again). The pen could also "drip" in the heat, which is why the inventor Henry Stephens devised a smash hit, in 1834, what he called a blue-black ink...

The ORIGINAL WRITING FLUID, which changes to a black colour, will be manufactured as heretofore.

...and advertised as "The Inks for Hot Weather". Please note: blue-black does not mean the actual color is between blue and black, but rather that writes in blue and turns into black as opposed to writing in brown and turning into black.

However, it was not long after this that people began writing in Prussian blues (a color that had been around for a while). It took a few years for the Stephens ink to make it to the United States (1837 or so), and in 1839 onward letters started regularly being written using a "Prussian Blue" mixture: instead of writing in blue and turning black these were pens that wrote and stayed blue. So while the Stephens invention was not meant to instill blue as a permanent color, writing blue as a color that stayed became normalized at the same time.

We also have an 1860 book (The History of Ink Including Its Etymology, Chemistry and Bibliography by Thaddeus Davids) which claims that blue started entering "common" use "ten or more years past" as opposed to specialized and/or decorative:

Inks of other colors than black were anciently used only for purposes of ornamental and decorative writing. In later and present times, red and blue inks have been extensively employed in ruling account-books and other paper for like uses. Blue ink, within ten or more years past, has been, with many, a preferred fluid for common writing.

The first Prussian blues were resistant to water, but starting in 1845 and especially through the 1850s there started to be blues that could run, including ones that could "be cleaned with soap and water". This leads to one other odd story, brought up by a turn-of-the-century expert on handwriting and ink (David Nunes Carvalho) who testified in many cases, including quite famously the Dreyfus Affair. In 1894 in New York -- the same year the Lexow Committee in the state senate exposed police corruption -- he advocated for a bill which introduced an official ink to prevent public records from being altered. Various inks were noted at the time to also clean with soap and water, and in the NYPD, they used one of the Prussian Blues, for "warrants and other public documents" where "the record can be easily altered". Allegedly, this use was intentional. By 1894 methylene blue ink had been around for a decade, with the additional property it was not easy to erase.

The bill did not pass. I'm unfortunately unclear if anything like it did, but that particular type of blue ink went away almost entirely by the 20th century.

...

For more on ancient methods of checking metal purity, see my answer on Ea-Nasir famously tried to sell sub-par copper ingot to Nanni, even though he promised fine quality ones. How would was the quality of copper ingot determined in c. 1750 BC Mesopotamia?

Ezzamel, M. (2005). Accounting for the activities of funerary temples: the intertwining of the sacred and the profane. Accounting and Business Research, 35(1), 29–51.

Hahn, C. M. (1981). The Topic is Ink. US Philatelic Classics Society, New York Chapter.

Krekel, C. (1999). The chemistry of historical iron gall inks. Int J Forensic Doc Exam, 5, 54-58.

Levinson, J. (2001). Questioned documents: A lawyer's handbook. Elsevier.