The Greek - and hence Roman - tradition is to list cases in the order: NOM - GEN etc. Dionysius Thrax (170-90 BCE) is considered to be the first extant record of this system - see a screenshot from Allen and Brink 1980, p. 65 The old order and the new: A case history (btw I strongly recommend this paper - imho it's the best summary of all relevant research on this question written in English):
cf. Jean Lallot (in Lallot 1985) argues this canonical order of the cases "certainly goes back to the Stoics" - cf. "L'ordre canonique des cas ... remonte très certement aux Stoiciens" (p. 54).
For those who read in Greek:
English translation (kindly translated by brianpck);
Here is a more precise account. There are three local relations: from a place, in a place, and to a place, which respectively are proper to the genitive, dative, and accusative cases. It is the case that everything moved desires to be moved around something immobile (e.g. a dancer cannot turn around if he does not place his foot upon the ground, which is immobile), and it is clear in the case of someone who is moved from one place to another that, first, he comes to be in a place, and thereby does he come to a place. The genitive and dative cases seem somewhat similar to the immobile ground. The relation "from a place" which comes to be from the genitive to a place, i.e. to the accusative, advances by means of the dative, which shows the relation "in a place." For this reason, the dative has been placed before the accusative.
Allen and Brink 1980 write that another tradition, i.e. Nom-Voc-Acc-Dat-Abl-Gen, goes back to the Danish historical linguist Rasmus Rask (1787-1832). His rationale was that the neuter forms were "the simplest" (hence the Nom-Voc-Acc block); thus, he gives the declension paradigms in the order neuter-masculine-feminine.
cf. Allen and Brink (p. 78):
Allen and Brink do mention some earlier historical evidence from the eighteenth century though:
- Nom-Voc-Dat-Acc-Abl-Gen in Versuch einer an der menschlichen Sprache abgebildeten Vernunftlehre (1781) by the German language teacher and grammarian Johann Werner Meiner (1723-1789) (Bernd Naumann describes this work as "a philosophical grammar which was the result of his life-long experience as a language teacher and thus better than most contemporary works on this subject" - see the entry on Meiner in Lexicon grammaticorum: Who's who in the history of world linguistics);
- Nom-Voc-Acc-Gen-Dat (by the Danish scholar N.L. Nissen)
Later, the Danish classical philologist (and politician) Johan Nicolai Madvig (1804-1886) adopted this case order in his Latin grammar. This caught on - via Kennedy, who used several grammars, including the one written by Madvig.
It is not quite correct to say that in German linguistics the canonical order is NGDA that is used to describe Latin declension paradigms. For example, the best Latin grammar in German (and it surely beats any Latin grammar written in English or French, imho), Lateinische Grammatik by Leumann , Hofmann and Szantyr (the last edition was published in 1977, edited by Manu Leumann) uses the following order of the cases in Latin:
Nom. Akk. Gen. Dat. Abl.
cf. Ausfürliche Grammatik by Kühner and Holzweissig (1912):
Nom. Gen. Dat. Acc. Voc. Abl.
In Russian linguistic research, the following order is the most common one: Nom - Gen - Dat - Acc - Abl - Voc. (really any Latin grammar).
In French linguistics, the following order seems to be common now:
N. Ac. G. D. Abl. (see e.g. Morphologie historique du latin by Alfred Ernout, 3rd ed. 1952; 4th ed. 2014).
So, the following order seems to be prevalent now in present-day Latin research (published in English, German, or French):
Nom. (Voc.) Acc. Gen. Dat. Abl.
Weiss 2009 (2nd corrected printing 2011), following the Indian system with some modifications, adopts a slightly different order though in his Outline of the historical and comparative grammar of Latin (and so does Sihler 1995):
Nom. Voc. Acc. Dat. Abl. Gen.
Pinkster 2015 (The Oxford Latin syntax) still uses the old order, Nom. Gen. Dat. Acc. Abl. Voc. but I suspect it's because Harm Pinkster is from the Netherlands.
I'll add more details tomorrow after classes.