just a few notes
1, Italy wishes) before 1936 (sanctions for ethiopia) Italy wished very much to become UK's affectionate southern cousin, even while Mussolini was in charge.
Even when it joined the triplice alliancein 1882, Italy imposed to add the clause: "The Royal italian Government declares that the provisions of the secret treaty [...] cannot [...] in any case be regarded as being directed against England".
[

note for any scotman/welsh irked by the UK<=>England : it is not my fault, they wrote England in the treaty

]
2, UK doesn't) On the other hand, UK often rebuffed there approaches, at first because being in its "splendid isolation", then because of its relationship with france, after 1936 because it began to sense italian navy as a potential rival in the med (they should not have worried).
The point that I am trying to make is that with gibiltrar-malta-cyprus-suez, UK already had its own mediterranean ports, thus it did not have to throw candies to a latin stranger to win his confidence, expecially if this meant being in worse terms with france.
Thus I thing that we need to find a POD in UK policy, rather than in italian's one
3, Athetism and religion) Sure, Savoia were atheists.
But most of italians were deeply catholics, and church authority was de facto far deeper than the king's one, even after 1870.
The fascist regime had so much support despite its obvious faults because managed to win the church approval with the 1928 concordate.
An official declaration of non-interest in religion in the royal house would have triggered a revolt.
The official conversion of an italian royal prince/princess to the protestant faith would have triggered a revolt.
Even just the marriage a royal prince/princess to a protestant princess/prince would have done Umberto/Vittorio Emmanuele throne quite shaky.
4, great war, UK point of view) UK was not angry with italy for its neutrality, since italy would have in principle to join the CP rather than the Entente: it was just by doing word-tricks and playing on the treaty clauses (such as the one I reported) that Italy managed not to join the CP.
Italy had to be bribed heavily (or at least had to be promised a lot) by London pact 1915 to join the Entente.
5, great war, Italian point of view) the great war did NOT cause significant disruption neither in italian economy nor in italian politics. Italy's industrial base was kept safe from the war and foreign investments (part of 1915 bribe) increased a lot.
6-month-long-governments were an endemic problem of the italian political system which pre-dated the war by alòong time.
Disoccupation rate in italy was very high and the war absorbed unemployed rather than taking away workers from tha factories.
Actually, tha real problem came when the war ended, since military expenses were to cut back and a lot of ws-soldiers become unemployed which were now also more prone to solve matters with violent means