Does anyone know which is the true first edition? The US or the UK? Thank you.
A quick Google search about the publishing dates for the first editions of the UK Version and the US Version reveals the US edition as the true first edition.
The date for the US Version was found by the Google Query - 'when was gilden fire US first edition released?' and the UK Version by - 'when was gilden fire UK first edition released?'
I tried searching for them as 'hardcover' as well, but the dates were no different from the general search. Feb 9, 2021 at 5:13
In any question about SF publications, your first stop should be the Internet Science Fiction Database. It includes publication details, and much else, for the vast majority of published SF for the past century.
The page for Gilden-Fire lists the first U.S. publication, in a limited 1000-copy (plus 275 signed copies) hardcover printing by Underwood-Miller, dated as 1981-11-00 (November 1981). Note that this listing is marked as "verified" which means that the listing details have been checked against a primary source, most likely either the publisher's records or a copy of the book.
The first publication in the U.K., by contrast, is a hardcover release by Collins dated 1983-10-00, or October 1983.
Note that if you're looking for a true first edition, which may be difficult to find with only 1275 copies, you should not confuse it with the Science Fiction Book Club (SFBC) edition (#2044) released later, which has the same cover art but a different background colour. It was still released, in October 1982, before the U.K. edition, though. (I know that buying older books secondhand online I've gotten burned a couple of times with SFBC editions.)