I think the Entente could go for minor gains if the Ottomans fought the Entente like the Germans did with the Entente and the Ottomans don't collapse so soon and if the Entente was suitably bloodied as well.
They did try, as in Gallipoli. In fact, until the empire simply fell apart in 1917, they did fight the Entente with as much courage, if with not as much equipment. And, in fact, the Turks in the end did manage to do more or less the equivalent.
Having said that, the other problem with the Straits is that Britain and Wilson wanted the Straits to be an international zone, rather than deal with the issues of a country having control over such a vital area, especially with a Communist Russia controlling the Black Sea. So that's another issue. However, I think that Britain would rather a Greek Constantinople than a Turkish Istanbul.