See the title: What if Admiral Milne, upon learning the Goeben and Breslau had passed through the Dardanelles either decided to, or was ordered to, steam into the Dardanelles and continue hunting Admiral Souchon's squadron?
I'm interested in the plausibility of this POD as well. Churchill had given authorization to engage before declaration of war, although that was shortly reversed, and the German and Ottoman story about "buying" Goeben and Breslau was a pretty flimsy justification, so it seems at least superficially plausible.
Tactically, could the Royal Navy Mediterranean units of 3 battlecruisers, 4 armored cruisers and other smaller ships "force the narrows"? The OTL attempt went fairly poorly (and included many more ships), but it happened after months of preparations that made it obvious to the Ottomans where the Entente would attack. I'm really unsure what the state of Ottoman defenses of the Dardanelles were in August 1914. They weren't closed to merchant shipping until weeks later, and I'm not sure about the significance of that fact. Did the Ottomons have mines they could remotely arm once they decided to close the straights? Or did Ottomans have to take time to lay minefields once they closed the straight? What was the state of Ottoman coastal artillery in August? If this POD results in a Royal Navy squadron of battlecruisers running headlong into a minefield and sinking catastrophically, it could have interesting ramifications for how the battle of the Heligoland Bight might be conducted in this ATL, because in OTL, it also featured battlecruisers operating close to enemy bases and minefields. How long could British units sustain operation in and around the Sea of Marmara? Also on the tactical level, I'm pretty confident the consolidated British Mediterranean fleet could destroy the Goeben, as well as the obsolete units of the Ottoman navy, but I'd be interested in hearing your input.
Strategically, what could happen next? Will this only bring the Ottomans into the war sooner, and what does that mean? Would the presence of a powerful British fleet in the harbor of Constantinople or cruising the Sea of Marmara cripple Ottoman war efforts, either through intimidation or by force? A loan Royal Navy squadron would need reinforcements if it forces the Narrows, but where do those reinforcements come from? Russia's Black Sea fleet is probably available to help, but how does Britain feel about Russia getting involved in controlling the straights?