True, Albion isn't that perfidious.
But I wonder if London might try to exert at least some influence by promoting some kind of an international (LoN-administered under only loose Russian control) arrangement for Constantinople and the Straits?
I don't think the Russian Empire would accept such an agreement, given their long-held ambition to control that region. Nor do I think that any LoN-equivalent would even exist in a scenario where the Triple Entente achieved victory without having to fall back on their large friend across the Atlantic.
Consider the impact of the Russian Revolution, assuming it still occurs. It could be a reasonable excuse for England to say that the deal was with the Czar, not the commies.
Russia would be pissed, but surely the rest of anti-bolshevik Europe (all of it) would be happier with the straights in Greek rather than Russian hands?
The main problem here is that the Russian Empire, pre-revolution, would want the Straits to be handed over to them, at least for them to have a troop presence there. I thought that the major goals of the Gallipoli campaign were (a) to open up another front for the British and French to fight the Central Powers from, (b) to cut off the Ottoman Empire from its German and Austro-Hungarian allies and (c), crucially for this purpose, to establish a direct connection between Russia and the Triple Entente. If the Straits are in British hands, the British will have a devil of a time preventing the Russians from getting there.
If there were then a successful communist revolution in Russia, the Turkish Straits would probably be a stronghold of the White forces, so the question is whether the RSFSR/USSR could be repelled from them by the Whites and their international backers. If there were such a revolution and if the RSFSR/USSR were repelled from the Straits, I agree that it would make sense for the Straits to go to Greece… but the two of those are rather big ifs.
And the potential impact of a successful Gallipoli is so great that the Russian Empire might well not be pushed to the point of revolution at all, and not just because of the huge morale boost of a victory over the Ottoman Empire, Russia's ancient enemy. People may discount the southern front of the First World War, but IOTL it played a very significant role in the collapse of the Central Powers once the tide was turned there.
Anaxagoras covered this possibility in a TL named, revealingly,
Rule Britannia!. Essentially, a successful Gallipoli means a short victorious war.
I recall reading in Martin Gilbert's (you may know him from hsi wonderful maps) book on the first world war that the British told the Russians they would give it to them even if the Russians did not put any men or ships into the attempt to seize it. And no real reason the British would give the land to the Greeks. The Romanians, Serbians, Bulgarians, and Greeks changed sides often enough depending on what king or junta was in charge.
I think you rather underestimate the Balkan nations. It's dangerous to say "X Country wanted Y" when X Country had plenty of important leaders who often disagreed with each other, but certainly some of the UK's leaders were very interested in securing the Balkan states' allegiance IOTL, especially that of Bulgaria. And the UK wanted Greece on its side so much that it more-or-less sponsored a
coup d'état against its own ideological preferences, deposing the Greek monarchy which it itself had helped to create and putting in its place an authoritarian Venizelist republic/quasi-republic which favoured joining the Entente! Greece wasn't a fickle power changing sides from neutrality to the Entente; its side was changed for it,
by the Entente. As for Serbia, Serbia hadn't changed sides for literally decades by the time of the First World War. It's true that Vienna
hoped that Serbia would turn Austrophilic once the Obrenovići were deposed by the Karadjordjevići (which has to be one of the most wrong analyses in history) but Vienna was utterly mistaken; the Karadjordjević dynasty was even more committed to opposing the Habsburg empire than its Obrenović predecessors were. I'd recommend Misha Glenny's
The Balkans, though it isn't exactly a small and concise book, as a good source for this.