19th century British and American authors are often discussed in the same vein of "Victorian literature".
Was it common for those authors (or their publishers/syndication firms) to publish their works in the other English-speaking countries at about the same time as they did locally?
Or did this happen years later and only involve the works that became bestsellers/classics?
Short answer:
British authors were much more popular in the United States than vice versa. To a great extent, their works were pirated rather than "published".
Discussion:
They were printed and sold, and had substantial audiences, but without any royalty or other payment to the authors until the latter part of the 19th century when copyright issues were worked out. Britain had a long literary tradition and prestige, the United States did not-- so there was an asymmetry, and for American publishers the prospect of a "free" British product from a well known author was attractive.
While both American and British copyright laws trace back to the "Statute of Anne" (aka "Copyright Act 1710"), the new United States in its first copyright laws of 1790 and 1830 restricted this protection to citizens and residents of the United States.
So American publishers reprinted British authors, paying them nothing. This was famously annoying to such figures as Charles Dickens, and you get the "British Authors Petition to Congress" in 1837 -- signed by dozens of leading writers. Among the people signing the petition are the [then future] Prime Minister Benjamin Disraeli who'd enjoyed considerable success as a novelist. Dickens travels to the United States in 1842, in part in the hope that he might stir up popular support for international copyright . . . he was disappointed in these hopes.
In the absence of an American copyright protection for the British authors they were printing, American publishers had to race each other to get their editions out the door and sold quickly: here the Philadelphia publisher Matthew Carey comments in 1823 on his plans to print "Quentin Durward" by the then best-selling Walter Scott
In two days we shall publish it here and in New York and the Pirates may print it as soon as they please. The opposition Edition will be out in about 48 hours after they have one of our Copies but we shall have complete and entire possession of every market in the Country for a short time
Note the irony -- Carey is here referring to other printers who might also print editions of the British authors he wasn't paying as "Pirates" (the use of that term for violations of intellectual property apparently dates to the 17th century).
It wasn't until much later that there was international copyright protection -- the US ultimately passed the "The International Copyright Act of 1891" and American authors didn't gain popularity in Britain for many years. Interestingly, Mark Twain visited England for the same reason that Dickens had come to the US, to obtain protection for his works -- at this point, in 1872, he could get a British copyright (at a time when British authors could not get an American one).
Sources:
Andrew J. Eaton. “The American Movement for International Copyright, 1837-60.” The Library Quarterly: Information, Community, Policy, vol. 15, no. 2, 1945, pp. 95–122.
Baetzhold, Howard G. “Mark Twain: England's Advocate.” American Literature, vol. 28, no. 3, 1956, pp. 328–346.
Bracha, Oren. “The Adventures of the Statute of Anne in the Land of Unlimited Possibilities: The Life of a Legal Transplant.” Berkeley Technology Law Journal, vol. 25, no. 3, 2010, pp. 1427–1473.