The title of the question more or less says it all.
The question asks for precise and scientific descriptions (submission-rules, editor-behavior, referee-recruitment, anonimity issues in an age where handwriting was still the norm, all the way to rejection or acceptance-and-concomitant-galley-proof-process), examples (maybe even scans of some handwritten referee reports from this journal and this time), or pointers to articles in historical journals on precisely this topic.
The motivation is mainly historical interest, together with my having to deliver a referee report posing some problems, and some hope on my part that even from seemingly-ancient examples one can learn new (or at least be remembered of known) useful ideas for the perennial problem of refereeing mathematical papers.
Remarks.
- It would not surprise me if (a question equivalent to) the present question had been asked recently on the web, but I did not find it (though not searching around for long).
- It would also not surprise me (see below) if there already was a book like "The Acta Mathematica—Then and Now", or something like that, in folio format and replete with high-resolution photographic reproductions. But I did not find one. There is a biography on Mittag-Leffler by A. Stubhaug. There is a book on letters between Poincaré and Mittag-Leffler by P. Nabonnand, but both seem not to contain information on the refereeing process of the Acta Mathematica in the early days, let alone a dedicated study.
- I decided against phrasing the questions in the form "How were mathematical referee reports written in the pre-typewriter-age?" since this would involve the ill-defined concept pre-typewriter-age. Typewriters gradually grew into being during the 1800s, notable hot-beds having been Italy and the USA. One could have phrased the question as "What are concrete, well-documented examples of non-typewritten mathematical referee reports?" but it seemed to me that localizing at the Acta Mathematica could be a more fruitful and focused historical question. This journal's history is likely to be very well documented. The late 1800s is a period which sits squarely within what is conventionally called modernity, and many original documents are bound to be extant even today, especially in a country as—relatively—untouched by the turmoils of the twentieth century as Sweden.
I chose the end-point 1918 of the time-interval asked about for no precise reason.
Most importantly, the journal is operational to this day, and there may be many Swedish mathematicians among MO users who are in the know, or even actively involved in the Acta Mathematica. I did decide upon asking it here—and not in some other, more historical forum—consciously.
Best Answer
Looking over the following references, I believe that they contain some of what is being asked after:
(Sample quotation: "The article examines the processes of refereeing, and the role of the editor, at nineteenth century mathematical journals by looking at the Cambridge Mathematical Journal and Acta Mathematica," p. 27.)
One of the references in the above manuscript is Chapter 8 (pp. 139-164) in an earlier work:
(Chapter 8 is entitled: "Gosta Mittag-Leffler and the Foundation and Administration of Acta Mathematica.")
And among the many references in this latter chapter is the following article of potential interest: