Solution 1:

It is hard to say for certain, but there could well be, yes.

As coconut.aminos’ answer quotes, the word first entered the English language from Spanish and/or Portuguese around 1620. The first citation in the OED article (paywalled) is from 1623, so we can probably assume that the word was first borrowed some time between 1600 and 1620 – let’s just assume it’s probably closer to the former for simplicity.

There is some evidence that this isn’t just a case of a word being fully Anglicised vs. an indigenous pronunciation being attempted as quoted in the other answer. I’ll expound (rather verbosely) on some of that evidence here. For a tl;dr, please see the bottom of this answer.

The primarily-British form /ˈʤʌntə/

The initial consonant
At the turn of the seventeenth century, there was in Spanish an ongoing simplification of the sibilant inventory: in both Old Spanish and Old Portuguese, junta would have been pronounced with an affricate /ʤ/ as [ˈʤunta] (as it still is in Italian today) or with a simple fricative as [ˈʒunta]. In Spanish, the affricate variant had all but disappeared by around 1500, leaving only /ʒ/. A similar reduction took place in Portuguese; I can’t find any sources that date the Portuguese change exactly, but I believe /ʤ/ lingered longer in Portuguese than in Spanish.

In Spanish (but not in Portuguese), the voiced /ʒ/ subsequently merged with unvoiced /ʃ/. When exactly this happened seems to be somewhat of a contentious point – some sources I’ve found say it happened already in the mid-sixteenth century, others that it happened in the seventeenth century. I don’t think we’re too far off if we assume that it had happened in some places by 1600, but hadn’t spread to the entire Hispanosphere.

Most likely, thus, English travellers who encountered and borrowed the word from Spanish and Portuguese speakers around the turn of the seventeenth century would have heard it spoken with a mixture of [ʤ], [ʒ] and [ʃ].

Now, of those three, /ʤ/ and /ʃ/ are very common phonemes in English, and both frequently appear at the beginning of words. In contrast, /ʒ/ is the rarest consonant phoneme in the language at all, occurring only in (mainly French) loan words, and notably it is almost entirely absent at the beginning of words. All words in English which have initial /ʒ/ are recent borrowings, dating no further back than around 1800 onwards: genre /ˈ(d)ʒɑːnrə ~ ˈʒɑ̃ːrə/ from the early 1800s; zho /ʒəʊ ~ zəʊ/ from around 1840; zhuzh/zhush (or zhoozh/zhoosh) /ʒʊʒ ~ ʒʊʃ/ from around 1970; etc.

In other words, to the English speaker who encountered [ˈʒunta] around 1600 (and probably also heard [ˈʤunta] here and there), /ʒ/ was likely not a phonotactically possible initial consonant. The obvious choice would then be to substitute the closest equivalent, which is /ʤ/ – also the consonant generally used for French loanwords at the time.

If these same people also encountered [ˈʃunta], the obvious choice would have been to borrow this as it was, since /ʃ/ was a perfectly valid consonant to start a word with – but the fact that there don’t seem to be any cases where the word junta can be shown to have been pronounced with a /ʃ/ rather speaks against this. It seems likely that either the devoicing of /ʒ/ to /ʃ/ in Spanish had not begun yet when the English speakers encountered the word, or it had not yet spread to the areas from whence they picked up the word.

The vowel(s)
At around 1600, it is likely that English still possessed a short, back, high, rounded vowel phoneme /u/ (see The Cambridge History of the English Language, vol. III: 87–91), whereas final, unstressed /a/ would probably have been reduced to [ə] already. The upshot of that is quite simple:

Spanish/Portuguese [ˈ(d)ʒunta] would be most likely to be borrowed as [ˈʤuntə] in English around 1600.

Later on, the /u/ phoneme split in twain in most English dialects, becoming variously /ʊ/ (as in put) or /ʌ/ (as in cut); in junta, it became the latter, so the current pronunciation of the form borrowed in the early seventeenth century is now /ˈʤʌntə/.

The primarily-American form /ˈhʊntə/

This form differs from the primarily-British form only in the initial consonant and the first vowel; the rest is identical. But as it happens, these first two sounds are quite enough to show that this form is a much later borrowing than the British form.

The consonant
As mentioned above, voiced /ʒ/ in Spanish eventually merged with /ʃ/. The merger was probably complete throughout the Hispanosphere by the middle of the seventeenth century at the latest. In Portuguese, this change did not occur, and /ʒ/ remains /ʒ/ to this day.

Old Spanish (like Old Portuguese and most other Romance languages in the area at the time) had developed seven different sibilant phonemes, including affricate sibilants. In Spanish, the affricates had lost their affrication, which led to some sibilants being distinguished only by marginal features, like apical /s̺/ contrasting with laminal /s̻/, both also contrasting with palato-alveolar /ʃ/. Phonetically, the apical variant is the frontmost of those three, and the palato-alveolar variant is the backmost.

Having three sibilants in such close proximity was apparently felt to be too indistinct, and a long process began whereby the sibilants were moved further apart in order for them to be more easily distinguishable.

The laminal phoneme /s̻/ historically came from /ts/ and /dz/ first losing their affrication and becoming laminal /s̻/ and /z̻/, and then, when voiced sibilants were unvoiced, merging in /s̻/. During the differentiation process, this frontmost phoneme moved even further forward towards the teeth and eventually became the interdental /θ/ sound found in Modern Spanish.

The apical phoneme /s̺/ was historically the original /s/ phoneme, as well as the result of the original /z/ phoneme losing its voicing. During the differentiation process, this phoneme did not change – in fact, it remains apical in Spanish to this day (unlike English /s/, which is laminal).

The palato-alveolar /ʃ/ was directly inherited from Old Spanish /ʃ/, as well as voiced /ʒ/ with loss of voicing. During the differentiation process, this backmost phoneme moved further back in the mouth eventually ending up moving so far back that it became a velar consonant and lost its sibilant quality, becoming /x/.¹ Importantly, this change did not happen until some time around the mid-seventeenth century (see this article, p. 173 mid).

The phoneme /x/ has been weakened in many forms of Spanish, and there is often more or less free variation between [h] and [x] as pronunciations of it. It tends to be ‘strongest’ (i.e., with most friction, most like [x]) in Spain and ‘weakest’ (with least friction, most like [h]) in Latin America.

English does not have /x/ as a native phoneme. The sound does occur in some loanwords (loch/lough from Scottish/Irish, Bach from both German and Welsh), even initially (chutzpah from Yiddish), but it remains a very rare sound of questionable phonemic status: it can usually be replaced by either /k/ (finally) or /h/ (initially).

In words of Spanish origin, Spanish /x/ is nearly always represented by English /h/. Names like Juárez and José or common nouns like jalapeño are not pronounced with [x] unless a particularly Spanish pronunciation is affected.

The initial consonant in junta /ˈhʊntə/ thus fits the typical pattern of a Spanish loan word containing the phoneme /x/ – but since the phoneme /x/ did not exist in Spanish yet when the word was first borrowed, the pronunciation /ˈhʊntə/ cannot represent the pronunciation that was originally borrowed: /ˈʤʌntə/ must be older than /ˈhʊntə/.

The vowel
As noted in The Cambridge History of the English Language linked to above, the vowel /ʊ/ (as in put) likely only became distinct from /ʌ/ some time during the seventeenth century, both originating in earlier /u/. Here it’s important to note that in the timeline given, it was /ʌ/ which diverged first – in other words, in three chronological stages:

1. Only one phoneme /u/ exists, pronounced [u]; cut and put are [kut] and [put]
2. /u/ is lowered and moves towards [ʌ] in some words, but remains [u] in other words. There are two phonemes, /ʌ/ and /u/; cut is now [kʌt], put remains [put]
3. The remaining cases of /u/ are centralised and move towards [ʊ]. There are two phonemes, /ʌ/ and /u/ (or /ʊ/), the latter pronounced [ʊ]; cut remains [kʌt], put is now [pʊt]

Loanwords where the source language has /u/ and English has /ʌ/ were likely borrowed at stage 1 of this. In stage 2, there is still an [u] sound in English, and it would be unexpected, if the source language and the target language both contain the same sound phonemically, if a different sound were selected. In stage 3, English no longer has an [u] sound, but [ʊ] is acoustically closer to [u] than [ʌ] is, and it would still be expected that the closest equivalent would be selected.

The pronunciation /ˈʤʌntə/ thus points to a borrowing at stage 1, whereas /ˈhʊntə/ points to a later borrowing at stage 2 or 3.

Complicating matters here is the fact that /u/ developed differently in different dialects – indeed, in some, primarily Northern English, dialects, they haven’t diverged at all, and both cut and put have [ʊ]. If the word was originally borrowed during stage 2 or 3 by speakers of such a dialect and then brought back to an English-speaking community where other speakers were used to inferring a difference between [ʌ] and [ʊ] from speech which had no such distinction, there is no real way to predict which variant would become the common one.

The difference in vowel quality between the two Modern English pronunciations is thus not solid evidence for an internal timeline, but they do lean towards /ˈʤʌntə/ being the earlier form, in support of what the initial consonant tells us.

Conclusion

There are several facts that need to be seen in light of each other here:

• /ˈʤʌntə/ appears, on phonological grounds, to be earlier than /ˈhʊntə/
• /ˈhʊntə/ must have been borrowed after the mid-seventeenth century
• /ˈhʊntə/ is most prevalent in the US
• reduction of /x/ to [h] in Spanish is most prevalent in Latin America

Dictionary.com give the following explanation about the pronunciation (also quoted in the other answer, here edited for IPA clarity):

The 20th century has seen the emergence and, especially in North America, the gradual predominance of the pronunciation /ˈhʊntə/, derived from Spanish /ˈhunta/ through reassociation with the word’s Spanish origins.

I cannot find any sources that deal with the pronunciation of this word before the twentieth century, so I have no reason to doubt that this is in fact true.

Based on these facts, I would consider this the most likely scenario:

Junta was originally borrowed around 1600 by English speakers who encountered it in Spanish and Portuguese. The word was Anglicised as much as any loanword, by approximating impossible phonemes with possible ones. This pronunciation yielded Modern English /ˈʤʌntə/.

Later on, perhaps even only in the twentieth century, exposure to Spanish became more widespread in the US, and it became obvious that there was a disconnect between the word’s English pronunciation and its obvious Spanish origin. This led to a gradual change in pronunciation favouring the Spanish pronunciation, helped along by the fact that the frequent reduction of /x/ to [h] in Latin American Spanish meant English speakers didn’t have to figure out what to do with an unusual initial [x] but could simply use the perfectly common [h].

This change, which is nigh ubiquitous in American English now, has not (yet?) gained much traction in British English where there is less of a Hispanophone presence.

^{1 This development, which may seem strange to an English speaker (going from /ʃ/ to /x/), has actually been recorded elsewhere as well: the Swedish sje-sound also derives from /ʃ/ but is now mostly some sort of velar-ish sound.}