Would Britain have taken the Dardanelles as a colony after a successful breakthrough?
Short answerNo; after the Allied promise of Constantinople, Russia would receive the capital while British and other Allied forces redeployed elsewhere.
NotesBritain would not simply convert the captured straits into a colony. In the 18 March branch, Britain and France had already promised Constantinople to Russia, so Russian forces would be expected to secure the capital. British, French, and imperial troops could then move toward the front against Austria-Hungary while shipping reopened the supply route. An operational victory would still be constrained by commitments already made within the Allied coalition.