Why is AD in Latin and BC is in English?

The 'BC' chronology is closely related to the notion of the negative numbers. Because 300 BC is actually the -300th year BC (well, actually -299th to be precise, but that's an unnecessary complexity). The idea of the negative axis going back to the negative infinity from zero is not that old, so, as was noted earlier, it relates to modern times when English was more common than Latin

It remains unclear at the moment exactly how and why, nor even exactly when, the dating system was named popularly in English language and then kept that way while the years after the year 1 in the Common Era retained their Latin abbreviation.

The timeframe to look into for a "first English usage of Before Christ abbreviated as BC" seems to be located in time only after the 17th century.

What can be ascertained is that the characteristics of the dating system itself developed in a curious way: 'what came after' Christ was named first, while what came before Christ was named so later.

It seems for the whole concept of 'dividing all time' at that specific point to have taken hundreds of years to even become established conceptually. With the sub-concept of 'Before Christ' itself, then still labeled in various forms of Latin words, only becoming popular after the 1470s, with any specific wording for the concept, including the literal native English "Before Christ/BC" even much later than that.

As can be seen on the Wikipedia pages:

The Venerable Bede

also helped popularize the practice of dating forward from the birth of Christ (Anno Domini – in the year of our Lord),

which was at roughly the same time made a thing in continental Europe through the decrees of Charlemagne.

As the idea to use exactly this intersection to count years in a 'new era' seems to have come from Exiguus around 500 (AD), we see the adoption of this system taking itself 'some time' until finally in 1422 Portugal also switched to this dating system officially.

It is sometimes said that with Bede's popularisation of the concept for AD (and the actual words used for it), it follows naturally that the matching concept of BC was invented at the same time. This seems not to be entirely true, as Bede indeed seems to have used the concept, but it didn't stick then.

Although Anno Domini was in widespread use by the 9th century, the term "Before Christ" (or its equivalent) did not become common until much later. Bede used the expression "anno […] ante incarnationem Dominicam" (in the year before the incarnation of the Lord) twice.

"Anno ante Christi nativitatem" (in the year before the birth of Christ) is found in 1474 in a work by a German monk. In 1627, the French Jesuit theologian Denis Pétau (Dionysius Petavius in Latin), with his work De doctrina temporum, popularized the usage ante Christum (Latin for "Before Christ") to mark years prior to AD.

Really, it is found quite often in the "German monk" Rolevinck's work. Hundreds of times, to be sure, although in the form of a mixture of Greek abbreviation/code and Latin:

Werner Rolevinck in Fasciculus temporum (1474) used Anno ante xpi nativitatem (in the year before the birth of Christ) for all years between creation and Jesus. "xpi" comes from the Greek χρ (chr) in visually Latin letters, together with the Latin ending -i, thus abbreviating Christi ("of Christ").

— WP: Anno Domini

This concept was apparently 'new'; or at least unusual and strange enough so that Rolevinck had to account for his audience with an explanation of the idea:

This unusual arrangement, with its inverted dates, in his opinion needed some clarification: '…ut eadem facilitate unico numero inspecto sciatur quot annis hoc aut illud factum sit ante nativitatem christi…' [so that seeing the date makes it easy to make out how many years before the birth of Christ something had happened].

— Johan Martens: "The Fasciculus Temporum of 1474 On form and content of the incunable", Quaerendo, Vol 22, NO 3, 1992.

Rolevinck was surely:

one of the first to make systematic use of BC dating.

— Mark A. Lotito: "The Reformation of Historical Thought", Brill, Leiden, Boston, 2019. p64.

Despite Rolevinck's Fasciculum being a popular book, printed and translated widely into Dutch, French and German, an early English equivalent of his texts still eludes me. If we then see that it took until in 1627 a French Jesuit Denis Pétau popularised the Latin ante Christum, as a direct equivalent of English 'Before Christ', we see that the usage of "AD" was by then well established, while the usage of anything for 'before that' saw a wide ranging choice of changing options, not always abbreviated but simply spelled out when the rare need to use such a date arose: AC, ACN, AAID, AChrN etc.

The third answer at Guardian is perhaps useful:

Anno Domini was first used by the church in England in the seventh century and came to be used in secular legal documents at a time when they, like church documents, were written in Latin. Dates `before Christ' were of little interest to the medieval church or to lawyers. The expression arose later when it was natural to use English. Cedric Hoptroff, Milton Keynes

This is supported by the same question on the English Language Stack exchange, with the following answer

Essentially the same information, presented in what may be a clearer manner:

The concept of dating years B.C. is fairly recent. The English abbreviation began during the Renaissance. Before that, medieval historians dealing with pre-Christ history usually referred to the reign of various kings and emperors.Deseret news

Having said that, I don't think there is a strong authoritative answer,

"Why the terminology changed from Latin to English is a matter of speculation." SydneyMorningHerald

(Aside: At least for me, the google search provides multiple answers; I just cherry picked the top results. I realize that the results of google searches vary by individual, and I also realize that the google search wasn't useful until we refined the question. Please note with emphasis that I am not saying the question was trivial; I think this is a valid function of H:SE - to help to reformulate questions into forms that other reference tools can engage with.)