Lay/lie may be the most commented on and written about usage item of all time.
First, I’m not aware of any regional distinction. As far as I know, the situation in Britain and North America are the same, although most of my sources on the subject are American.
As usual, Merriam-Webster’s Dictionary of English Usage gives the most thorough review.
The intransitive lay has been going strong since the late thirteenth century. (e.g., from the late 13C poem The Harrowing of Hell: “Sathanas, y bynde the, her shalt thou lay.") Then, as is often the case, eighteenth century grammarians started to declare it incorrect for no particular reason. In this case, the grammarians were rather successful at driving it out of written use, but it has remained very common in speech. M-W notes that it is not a reliable social marker—those who avoid it in writing commonly use it in speech—even though some commentators and writers try to use it as class indicator.
In the late twentieth century, prescriptivists invented another distinction, that the intransitive lay may be used for objects, e.g., “the book lays flat,” while lie should be used for people. Again, this distinction does not seem to be made in actual usage.
Also, in the late twentieth century, the intransitive lay started to appear more in published writing, indicating a loosening of the shibboleth. But it’s still not completely accepted by copy editors. Most grammarians, even if they like the distinction, have declared that attempting to maintain it is futile. Garner rates it a 4 on scale of language change, “virtually universal but is opposed on cogent grounds by a few linguistic stalwarts (die-hard snoots).” (The difference between stage 4 and stage 5 appears to be Garner’s idiosyncratic preference. If he personally observes a widely ignored rule, he and those like him are “linguistic stalwarts” and it rates a 4. If he doesn’t, those who insist on the distinction are “pseudo-snoot eccentrics” and it rates a 5. In Garner’s parlance, being a snoot is a praiseworthy thing.)
The intransitive lay is also used in nautical speech, where it has always been uncontroversial and considered standard, e.g., “lay at anchor.”