I don't see how they could have come into the public domain in the UK, since there copyrights are measured from death dates-- AP Carter (in whose sole name all the Carter Family copyrights were registered) lived until 1960--add 70 years till expiration?

Furthermore, I think the international conventions extend copyrights to the later of the date per one's national laws and the date per the laws of the original nation of copyright, so any continuing copyright in the US would affect the UK as well (just as in-effect UK copyrights begun prior to 1923 prevent free usage in the US.)

But that's just copyrights on the songs. The terms for recording copyrights are quite different (though as the songs themselves--in the Carter arrangements--would still be copyrighted, wouldn't one have to wrestle with both types of copyright in order to reproduce a recording?)

As for the tunes, many of the "Carter Family" songs were not in fact by the Carter family, as this and Ritchie's other Carter thread amply demonstrate, so you may record and use those tunes with impugnity, with no credit due to the Carters whatsoever, provided you can work from earlier sources (frequently cited in these threads).