@jamesc27, Dean Martin's version of "In The Cool, Cool, Cool Of The Evening" was released on Capitol CL-13575, but it doesn't look like Rosemary Clooney's had a UK release. According to the book "First Hits", the only other UK issues of the song were by Bing Crosby & Jane Wyman (Brunswick 04760), Art Lund (MGM 438) and The Five Smith Brothers (Parlophone R-3440).
The Rounders is here a pseudonym for Harry Reser's band with vocalist Scrappy Lambert (mx 8985-2 recorded 30 Aug 1929)
La Palina Broadcasters is here a pseudonym for Fred Rich's band with vocalist Smith Ballew (mx 9009-2 recorded late Aug 1929)
Source: Brian Rust's American Dance Band Discography
Some copies of this apparently contain a different recording on the A side - the same tune but by Len Fillis & His Hawaiian Orchestra with vocal by Al Bowlly (mx 1718-2). My copy however contains the California Ramblers version matching the matrix number (3857).
There also appears to be a third version of the A side on some copies - a non-vocal recording by Cecil & Leslie Norman's studio band (see here).