Monarchist countries like the Netherlands, Belgium, Denmark, Norway, Sweden, the UK ... that either stayed out of the war mess or ended up on the winning side were more likely to keep their monarchy.
Prior to the military conflagrations that started in 1914, vast majority of Europeans lived in monarchies, even though the democratization and personal emancipation process was well underway. The only republican countries in Europe at the outbreak of WWI were France, Switzerland and (very recently) Portugal. Well, and San Marino, yes.
If I'm not mistaken, those are also countries that had already significantly diminished the actual power of their monarchs prior to WWI. The UK in particular arguably started that process centuries earlier with Magna Carta. On the other hand, places where the monarch retained the most power, like Russia, saw their monarchs assassinated pretty early in the industrial period.
Today, 80% are still supporting monarchy [1], also among the young [2]
[1] https://www.nrk.no/norge/8-av-10-vil-beholde-kongehuset_-sto...
[2] https://www.nrk.no/norge/3-av-4-unge-stottar-opp-om-monarkie...
I bet that is exactly would have happened. The reality is that the advance of science and technology and general literacy was quickly eroding the traditional notion of the monarch having the consent of God or heaven.
By the turn of the 20th century this was almost complete, and largely irreversible. From that point on, the only value left in the monarch in industrialized societies was as a symbol of a national ethnic identity for societies that defined themselves in those terms.
Today, only the most extreme religious movements around the world see a return to monarchical society as a goal. Even the neo-fascists would prefer a dictator drawn from their ilk, not a king.