When and why did “mess” come to mean an untidy condition?

The answer lies in convergent evolution. These are not the same word, and merit two distinct entries in any dictionary worthy of that name.

The version of mess that derives from Latin mittere meaning to send or put (think English missive, mission, Mass) is today used mostly in a military context, but was once also used for Roman Catholic rite. The Wikipedia article says:

The root of mess is the Old French mes, "portion of food" (cf. modern French mets), drawn from the Latin verb mittere, meaning "to send" and "to put" (cf. modern French mettre), the original sense being "a course of a meal put on the table"; cfr. also the modern Italian portata with the same meaning, past participle of portare, to bring. This sense of mess, which appeared in English in the 13th century, was often used for cooked or liquid dishes in particular, as in the "mess of pottage" (porridge or soup). By the 15th century, a group of people who ate together were also called a mess, and it is this sense that persists in the "mess halls" of the modern military.

In contrast, the other word mess for, as you put it, an untidy condition, is more likely related to the verb to muss as in mussing up someone’s hair and may according to Wiktionary be a corruption of a Middle English verb mesh meaning to mash.

A recently updated entry in OED Online [OED Third Edition (September 2001) - fully updated; OED Online version September 2016] does not distinguish the etymology of 'mess' in the sense of

I. A portion of food, and related senses.
1. a. A serving of food; a course; a meal; a prepared dish of a specified kind of food. Also fig. Now hist. and Eng. regional (except as merging into sense 2a).

^{Figurative uses of this sense (for example, quots. 1570, a1764) are often indistinguishable from the more pejorative senses 2c and 3a.}

["mess, n.1". OED Online. September 2016. Oxford University Press. http://www.oed.com/view/Entry/117092?rskey=lnyOAS&result=1&isAdvanced=false (accessed September 18, 2016).]

from the etymology of 'mess' in the sense of

2. c. An unappetizing, unpalatable, or disgusting dish or concoction; an ill-assorted mixture of any kind, a hotchpotch.

(op. cit.)

or from 'mess' in the sense of

3. a. fig. A situation or state of affairs that is confused or presents numerous difficulties; a troubled or embarrassed state or condition; a predicament.

(op. cit.)

For all three senses (and others, including "3. b. A dirty or untidy state of things or of a place; a collection of disordered things, producing such a state" [op. cit.]), the etymology given is

Etymology: < Anglo-Norman mes, mees, messe, Old French mes portion of food (mid 12th cent.; Middle French, French mets dish, food) < post-classical Latin missus portion of food, course of a meal (4th cent.), spec. use of classical Latin missus, lit. ‘sending’....

(op. cit.)

Notwithstanding a variety of more or less poorly-considered folk etymologies, the derivation of the senses of 'mess' is not to my knowledge controversial. Senses I3a and I3b, for example, developed from figurative use of sense I2c, which itself reflects pejorative uses of sense I1a.