Neither McKay, nor Scott, nor Whatmore mention any serialization of LOVE ETERNAL.
McKay, Scott, and Whatmore claim that Cassell and Company published the first edition in London on April 4, 1918. This first edition includes a color frontispiece by A.C. Michael. The are no ads at the end of the book, but the last page is date-coded "318" (March 1918). McKay and Scott claim that 12,500 copies were printed.
Longmans, Green and Company published the first American edition on June 1, 1918. McKay and Scott both agree on the publication date in New York. Longmans claims it published this novel in May 1918 (see below). The American edition has no illustrations. The ads at the end of the book are not dated.
See the VisualHaggard.org page dedicated to the illustrations found in various editions.
I do not own any of the books pictured below, unless noted.