An awful lot is going to depend on exactly
how the wars don't happen, but doing a rough run down (and ignoring the likes of Monaco or Liechtenstein):
Most Likely to survive
I'd say that even surviving historically isn't a guarantee of surviving here- the lack of counter-examples/threats might encourage a more reactionary take for example- but anyone who survived the 20th Century historically has to be considered to have the best chance of surviving:
-Belgium, with the caveat that the handling of divisions between the Flemish and Walloons could doom it regardless.
-Denmark
-Luxembourg
-Netherlands
-Norway
-Sweden
-United Kingdom, with the possibility that Ireland ends up a Commonwealth Realm without WWI (and that's a
whole other topic)
I'd also put the following into this category:
-Bulgaria: Tsar Simeon, or at least the
idea of Tsar Simeon, was popular enough historically to get elected Prime Minnister after the fall of communism. Quite possible that if he's actually ruling Bulgaria for the second half of the 20th Century he ends up losing the throne, but this feels like one where we need to actively posit 'the monarch just completely screws up'.
-Romania: There was a similar popularity in post-communist Romania for
the idea of King Michael- neither case anywhere near the prospect of an actual restoration, but (while I'd defer to
@Yokai Man here) again it feels like Romania probably would have stayed a monarchy if left to itself unless the monarch screws up.
We can't really use the OTL personalities of either Michael or Simeon mind you considering how much they were shaped by exile and the Communist occupation of their homelands.
Likely to survive
-Italy: It's dependent on what happens in the 20s. A lot. No WWI possibly means that fascism doesn't get going, but desire for territorial aggrandisement in the Med is still going to be a thing. Italy only voted relatively narrowly for a republic in 1946 anyway (54-46 roughly) so it feels like it's got a good shot, but Italy also feels like the historic trends point towards 'the monarchy discredits itself through stupidity or backing the political reactionaries more than for somewhere like the Netherlands. There's a decent argument that it's as likely to survive as Belgium's, and the monarchy in the latter has just been lucky/managed to play their cards right.
-Spain: But wait, I hear you cry, Spain survived historically. Yeah and that's basically the only reason this is this high. There were
deep structural issues in the Kingdom in the early 20th Century- the establishment of the 2nd Republic feels more likely than not- and while it's still possible that the Spanish monarchy
could be restored regardless of the exact sequence of events that happened historically, it feels equally plausible that Spain just sees a series of republics of different stripes rather than ever bringing back the monarch-
@Gorro Rubio your thoughts here?
Potentially an option
-Germany: Absent WWI a lot of the reasons for the 1919 revolution are removed. On the other hand if the Savoys were willing to endorse a reactionary government to maintain their position, the Hohenzollerns were quite happy to try imposing one. Really without WWI there's probably some sort of crisis brewing in the 20s or 30s as Kaiser Wilhelm had all the political sense of a melon.
If it can survive him, and
if the Crown Prince not looking for a way to get the throne back means he's willing to be more moderate than historically- then either of his sons look like they could have been the sort of moderniser needed to ensure the monarchy's survival- though Wilhelm's historic decision to pull an Edward VIII seems like something that could have happened regardless. I'm not going to touch the constituent monarchies.
-Greece: Greece's monarchs really weren't the brightest or most capable bunch historically,
but a lot of the reasons for its downfall stem from events such as the National Schism (on whether to enter WWI), the disastrous Asia Minor campaigns after the Treaty of Sevres, the instability of the second republic and resultant poor handling of the depression and the growth of the military's influence on politics after this. It's still very possible that things go wrong- the 30s seem like a time when things could still have essentially imploded for the monarchy- but Greece feels like it's got a decent shot.
-Malta: Either within the UK or as a Commonwealth realm, either seem like something that just didn't quite work out historically.
-Montenegro: Will live or die on how tempting pan-Yugoslavism is. There's always going to be a faction calling for union with Serbia, and the better Serbia is doing the more likely it is they get integrated (though potentially as a federal sub-monarchy). Serbia and/or Yugoslavia seem more likely than not to go to hell in a handbasket however, so Montenegro's 'just stay quiet and don't get involved' tactics could be appealing. Perversely, the lack of WWI might mean that they end up being kept out of Serbia's orbit by a strong Italian alliance which may, or may not, doom the monarchy regardless.
-Russia: Has massive issues, and Nicholas II is an absolute idiot. This is one of those make-or-break situations however- most of the Romanov men seem to have died about 60, give or take, so if Nicholas makes it through the 20s as a reactionary, but gradually industrialising, monarch before dying it looks like the next heirs would be at least
willing to work with the Duma. High chance of the monarchy getting deposed in the 50s/60s IMO.
Unlikely to surivive
-Austria: NOT Austria-Hungary (for which see below). The loss of prestige from the almost-certain collapse of Austria-Hungary will be a significant issue for the House of Habsburg,
but if its gradual/managed enough they might be able to focus in on just Austria-Bohemia-Carinola. It's a big ask and it's going to take a lot of careful navigation, but there's a path forward for them.
-Portugal: The Estado Novo considered restoring the Portuguese monarchy to secure legitimacy historically. There's a possibility that they're able to achieve the Juan Carlos option but it's slim.
-Serbia: Serbia is going to manage fine for a bit. Then Austria-Hungary collapses and inevitably they're going to get mixed up in efforts to take Bosnia-Herzegovina/union with Croatia/Yugoslav ideals and the monarchy is going to get involved heavily here and basically it's just going to be a horrific mess no matter what happens. There is no objective reason why the Serbian monarchy
can't survive, but good grief they are going to have to roll sixes all the time.
No chance at all
-Austria-Hungary: As
@David Flin can expound upon at length, A-H was doomed, doomed, doomed. It's been on life support since the Ausgleich of 1866 and the annexation of Bosnia in 1908 just makes things worse. Franz Ferdinand's Federal Empire is a nice, but completely unworkable, idea along the same lines as the Imperial Federation for the British Empire.
-Iceland: Voted to break the Personal Union with Denmark by 98%.
-Ottoman Empire: I can't see the Ottoman Empire surviving past the 40s, and whereas the Habsburgs
might be able to reinvent themselves, the most likely end for the Ottomans- taken apart by foreign empires or torn apart from internal revolution and rebellion- looks more likely to spell a definitive end for the monarchy.
