Is the change from "m" to "n" within the descent from O.E."æmette" to E."ant" a regular one?

Solution 1:

The consonant change from emmet to ant is known as assimilation.

The consonant [m] is both bilabial and nasal and [t] is neither. This makes the consonant cluster [mt] difficult to pronounce because you have to change both features simultaneously (i.e. open your lips at exactly the same moment as you redirect the flow of air from your nose to your mouth). It is easier to change these features one at a time, resulting in the intrusion of a [p] (leading to [mpt]), or else drop the bilabial feature altogether (leading to [nt]). You can see from etymonline that both of these simplifications have happened at different places and times.

Here's John C. Wells in Accents of English (page 96):

A way of reducing the articulatory complexity of strings of consonsants is through assimilation, the process whereby a sound is made phonetically similar to the sounds constituting its phonetic environment. We know, for instance, that the word which is nowadays ant /ænt/ had an Old English form æ̅mete; regular vowel developments would give a present-day form *amt /æmt/ (compare traditional-dialect emmet). The change from [m] to [n] before following [t] is an assimilation which results from an obvious articulatory simplification, namely the elimination of a labial movement.

Some other words which have undergone assimilation include:

• accurate, affirm, etc., from the Latin prefix ad–.
• embody, embolden, etc., from the French prefix en–.
• impossible, impartial, etc., from the Latin prefix in–.
• comfortable, composition, etc., from the Latin prefix con–.
• symphony, sympathy, etc., from the Greek prefix syn–.

Solution 2:

It seems generally agreed that the word "ant" comes from Old English ǣmete. Your second source about the history of the word "ant" says that this word also took the form "ampte" in Middle English, but later changed further to "ant."

As Gareth Rees explains in his answer, the change of m to n was motivated by the process of assimilation. It's common for nasal consonants such as /m/ to assimilate to following stop consonants such as /t/. In fact, the same change happened in the history of the word aunt, which comes from Latin amita via Old French ante.

Even though the assimilation /mt/ > /nt/ does not usually occur in modern English, there are some comparable assimilatory processes. The consonant /n/ is commonly assimilated to /ŋ/ before velar stops, or to /m/ before bilabial stops. And according to the following paper, even /m/ and /ŋ/ are occasionally assimilated in some cases in modern English speech: "Assimilation of word-final nasals to following word-initial place of articulation in UK English".

Is the change of Old English m(..)t to Modern English nt regular?

I actually don't know if the assimilation of /m/ to /n/ in ant (from Old English ǣmete) should be classified as regular or irregular. On the one hand, the Oxford English Dictionary helpfully references another word with this sound change: scant, which apparently comes from Old Norse skamt.

But on the other hand, there are a couple of counterexamples where /m/ was retained before /t/ (with epenthetic "p" between them in the modern spelling, and optionally in the pronunciation). First, the word empty, which descends from Old English ǣmtig, ǣmetig. Second, the place name Hampton, which corresponds to Old English Hāmtūn.

Unfortunately, I haven't been able to find any other examples, as this sequence of sounds was not very common in Old English. So, I can't provide a firm conclusion.

One word that shows a similar assimilation is lenten, Lent, which the Oxford English Dictionary says comes from Old English lęncten (which presumably would have been pronounced with the velar nasal [ŋ]).

Similar developments at other times

There are a few similar words that developed clusters like these more recently in English. For example, Peter Shor mentions the contraction ain't, derived from am + not. Of course, not already starts with /n/, so this word isn't entirely comparable to ant. On the other hand, the words "dreamt" and "unkempt" have /m(p)t/ from contracted m followed by t.

There are also words that underwent this kind of change before entering English, such as the aforementioned aunt, the verb count (from Old French cunter, conter, which developed from earlier Latin computare) and the noun prince, which comes (through French) from the Latin word princeps, which already showed assimilation in Classical times (it is composed of the roots prim- as in primus "first, prime" and ceps from the verb capere "to take").