Wilson will be involved in any case, due to the US being a Great Power and holding a lot of Entente debt, what will change will be the level of influence he will have, plus the presence of another player will cause more negotiation.
A lower level of Wilsonian influence - means that Entente Agreements are more likely to stand unchallenged.
And Russia wanted the control of the strait...thing that nobody in France or Great Britain desire.
I don't believe France & Britain cared about the straits in particular - it was Russia taking over the entire Ottoman Empire (a nineteenth century fear) -- and or an uncontrolled collapse of the Ottoman Empire that they wanted to avoid.
This would have been an agreed division of the Turkey between the Entente Powers - with main beneficiaries Russia, France & Britain getting the bits they want and Italy, Greece and potentially Bulgaria (if they came in on the Entente side in September 1915) getting Thrace.
I don't think France & Britain would have cared a jot about the Straits.
The region of Smyrne was promised/assigned to Italy, plus there is the dodecanese question and in any case both Paris and London in OTL considered Italy getting a foothold in ME not a good thing and tried to limit their 'acquisition', this was one of the reason in supporting Greece claims.
And how Greece can be stopped? In the same way of Italy...holding back loans and witholding political and material support.
Treaty of London only gave Italy Anatalya Province in Turkey -- not Smyrna?
And I don't see why the entente powers would stop Greece - they didn't in the OTL.
Also remember that the Sykes Picot Agreement was only partly carried out in the OTL -- it assigned much of Turkey/Anatolia to Russia & France.
France after their invasion did make a half hearted foray into south eastern Turkey - but didn't stay.
But with Russian 'feet on the ground' in Armenia AND with the Russians raising hell over the Genocide....the French would go in.
The Tsar would shout about genocide to the hills to discredit the Turks and justify their presence.
And if a war weary France & Russia were facing Ataturk's Nationalist resistance - having Greece in (to do most of the Fighting) would be quite useful.
As for Italy if stayed or went - probably wouldn't make a huge difference - I suspect they would stay - they did briefly land troops in Anatalya - it was only the prospect of fighting a major war there that led them to pull out