It wouldn't surprise if the tutors made their way there with the proliferation of the banjo and American acts. Now, did they set up a table and sell merchandise? I highly doubt that. Lol...The way these things go, I imagine the tutors were sort of sprinkled in, given to other English players by some American players. But I'd guess the mass publication for distribution in England took quite some time, due to modes of travel and communications then...
That is what I was thinking.
I'm just making a start on the 1860 'Buckley's New Banjo Method' on my flush fret banjo, probably English and possibly made in the 1870s. I like the thought that it was used to play those tunes back then.
From what I understand, Sweeny was in England around 1848-ish. So, I'd guess it may have started around then, though it's well known he basically learned "by ear" from Uncle Eph. But I'd guess it was the groups that came not too much later that the tutors started to make their appearance. But, those groups had to be willing to get away from the "minstrel" costumes and such, as the English crowd couldn't identify with that portion of American culture at the time.
Sweeney was in the UK starting in 1843. Sheet music of tunes from the Virginia Minstrels was available very soon after they started up...but it was vocal music, songs. I would hazard to guess that copies of The Howe Preceptor and The Briggs Banjo Book (and all the rest) made it across within weeks of their publication. Sheet music sales was a hot prospect on both sides of the pond and there were no restrictions (or copyright agreements). Shipping across the pond via sailing ship might have been faster than current USPS...
If we set 1843 for the onset of the Minstrel fad in the UK, banjo music would have been a very hot seller 8 yrs later when the Briggs book came out.
To be historically accurate in accord with the first print evidence Mr Sweeney was in Liverpool and performing a year earlier in April 1842 (see The Liverpool Mail Tuesday 5 April 1842). Liverpool was one of the main Atlantic seaports and points of entry for trans-Atlantic passengers and almost certainly Sweeney's point of disembarkation in England. Certainly Liverpool was the venue of his first public performance in England.
It's clear from the content of at least one early English banjo tutor that the author(s) had come into contact with American banjo tutors, more specifically the Howe and Briggs publications. Other examples of early English banjo tutors are structured quite differently and have more in common stylistically with English contemporary guitar tutors.
The mid-19th century Atlantic shipping routes were frenetically busy. Migration was not always one-way; for a variety of reasons many people criss-crossed the Atlantic. My own family includes 19th century migrants who went to the US and then returned to England. There was wide scope for access to American material published the large port cities (New York, Boston, and Philadelphia) to an English readership.
There is no doubt that the arrival of touring American minstrel troupes in England popularised the banjo to English audiences. However, with the prevalence of black sailors on English ships crew lists it's far less certain that Sweeney was the very first occasion English people, particularly inhabitants of English seaports, observed or heard a banjo.
One purely anecdotal attempt to answer your question about how long after publication American tutor books were seen in England...
Cecil Hicks who wrote the first English banjo tutor book that we have found so far, was a professional musician and lived in London. One might expect him to have a slightly increased awareness of and access to any banjo publications that were available. In the introduction to his own banjo tutor in 1852, he claimed it was the first book of it's kind in the world. That means that two years after the Howe tutor was published in the US he had no knowledge of it.
The obvious plagiarism evident in one or two of the English banjo tutors may be a clue that copies of US banjo tutors were not on retail sale in England. The opportunity for direct contemporary comparison in a commercial context would potentially leave the British author(s) liable to penalty under rudimentary copyright law. I've seen no evidence in the period 1850 -1865 that the earliest US banjo tutors were advertised for retail sale in England.