I can only find the Second World War reference in post-WW2 editions. The original 1927 editions has

Such details are too horrible to dwell upon to-day. It was war carried to its utmost extent — an extent that was very nearly approached in the last European war. It was the slaughter of human beings without hatred — simply to make an end of them.

and the post-WW2 editions with "Second World War" have simply substituted that for "last European war".

Clearly, Lamb was referring to the First World War, writing in the 1920s. It looks like an editor decided that "last European War" might confuse post-WW2 readers; perhaps they also felt that the destructive brutality of the Second World War made a better analogy. Lamb was discussing the destruction of cities in Khwarezmia by the Mongols, with enslavement and massacre of civilian populations, and in the case of Urgench, the physical destruction of the city including diversion of the river to flow over the site. His "slaughter without hatred" suggests that his intended analogy was "industrialised killing", while the editor's substitution might have intended deliberate killing of civilians in WW2 as the analogy.

Lamb's Genghis Khan is an interesting book. Very engaging compared to a lot of historical writing, and very sympathetic towards a figure more often treated as somewhat of a bogeyman. The book comes across very much as if written by a skilled writer of heroic historical fiction - something that Lamb had been doing professionally for a decade before he wrote Genghis Khan. Genghis Khan was very successful, and IIRC, award-winning. It contributed to Lamb's Guggenheim Fellowship (to work on the history of the Crusades, including studying sources in the Vatican library and travel in the area of the Crusades).

Lamb's sources for this book are still the main sources for the topic: the Secret History and the Persian histories. Lamb makes use of the numbers for sizes of armies and massacres from the Persian sources rather uncritically (something later historians have also done), and the book is perhaps light on the Mongol-Jin wars. Especially today, it isn't the definitive history of Genghis Khan, but IMO it still makes an excellently readable introduction. If it inspires the reader to explore more recent historical work on Genghis Khan and his times and conquests, it continues to a do a fine job it has been doing for many, many decades.

References:

The quote from the 1928 Thornton Butterworth edition, 2nd printing, but it appears to be the same in all of the printings of the Thornton Butterworth edition, into the 1930s.

More on Lamb: http://www.haroldlamb.com/

The Guggenheim Foundation on Lamb: https://www.gf.org/fellows/all-fellows/harold-lamb/