Say the ships alone go ahead to ride or die, and the Ottoman government skips town for inland.
Inland to where? Inland to Thrace in Europe? Edirne/Adrianople? Or inland in Anatolia to Bursa or even Ankara?
Flight of govt leadership, and ship bombardments and fires would lead to mass panics and civilian flight from the city, making for a mess and prestige hit for Ottoman authorities.
The Ottoman authorities at that point could do a suing for peace in general, or fight on from their new inland capital.
If suing for peace in general, are they just saying unconditionally, or seeking terms, and likely to try to hold old for peace terms on a territorial status quo ante bellum basis (pre-war borders) or a territorial uti possedetis basis (everybody keeps what they got right now) or a set of extensive territorial concessions to the Entente based on what all expected the Entente could easily or inevitably seize? [the battle lines at this stage of 1915 had the Ottomans in control of most of their pre-war territory, in the Palestine theater, actually in occupation of some Egyptian Sinai, having only lost very small bits of land in northeast Anatolia along the border, and some of southeastern Iraq north of Basra and Kuwait]
If the Ottomans either seek general terms, or do not, but continue resistance for the time being, just from an inland capital, they could leave Constantinople bombarded and practically defenseless except for whatever pitiful batteries get suppressed there quickly by ship fire. Or, they could proclaim Constantinople an open city, to try to get it spared bombardment from that point on.
Continued bombardment and destruction of crowded Constantinople quarters when it's clear the city is defenseless could become a PR problem for Britain and the Entente, not just a martial embarrassment for the Ottomans, especially if the British Fleet continues with a destructive bombardment of residential quarters after an Ottoman proclamation of it as an "open city".
Of course the British might be a couple weeks from having sizable, self-sufficient landing parties at hand to take occupation of even an open city, because there will be plenty of infantry and police around.
Things could get messy, if the Ottomans, as a ruse of war, accepted a naval landing party from ships then bumrushed it to take hostages.
If the Ottoman government is running things from an inland capital, and the ships are dominating the straits, but it is still taking many weeks to assemble an adequate landing party to secure the shores of the straits, would the Ottomans in either Thrace or Anatolia or both use the time to rush concentrate artillery into coastal positions against the ships of the fleet and start striking them?