And there are equally as many American accents.

https://www.bbc.com/culture/article/20180207-how-americans-preserved-british-english

One feature of most American English is what linguists call ‘rhoticity’, or the pronunciation of ‘r’ in words like ‘card’ and ‘water’. It turns out that Brits in the 1600s, like modern-day Americans, largely pronounced all their Rs. Marisa Brook researches language variation at Canada’s University of Victoria. “Many of those immigrants came from parts of the British Isles where non-rhoticity hadn’t yet spread,” she says of the early colonists. “The change towards standard non-rhoticity in southern England was just beginning at the time the colonies became the United States.”

American actors have a head start with performing in OP: it’s “so much more American” than the prestigious Received Pronunciation accent in which Shakespeare’s plays are generally performed now, says Paul Meier, theatre professor emeritus at Kansas State University and a dialect coach who’s worked on theatre productions like an OP version of A Midsummer Night’s Dream.

For instance, Americans are already used to pronouncing ‘fire’ as ‘fi-er’ rather than ‘fi-yah’, as most Brits would.

It’s useful to know how words would have been pronounced centuries ago because it changes our appreciation of the texts. Because British English pronunciations have changed so much since the era of Queen Elizabeth I, we’ve rather lost touch with what Early Modern English would have sounded like at the time. Some of the puns and rhyme schemes of Shakespeare’s day no longer work in contemporary British English. ‘Love’ and ‘prove’ is just one pair of examples; in the 1600s, the latter would have sounded more like the former. The Great Vowel Shift that ended soon after Shakespeare’s time is one reason that English spellings and pronunciations can be so inconsistent now.

So what’s popularly believed to be the classic British English accent isn’t actually so classic. In fact, British accents have undergone more change in the last few centuries than American accents have – partly because London, and its orbit of influence, was historically at the forefront of linguistic change in English.

As a result, although there are plenty of variations, modern American pronunciation is generally more akin to at least the 18th-Century British kind than modern British pronunciation. Shakespearean English, this isn’t. But the English of Samuel Johnson and Daniel Defoe? We’re getting a bit warmer.

Are they all distinct accents, or are they slight variations on an accent?

Well, they're all equally fancy to me!