As for the Titanic herself, her career TTL would probably go along these lines: continues on the Southampton-NY run until WW1 breaks out. During WW1, if the British government decides not to have her continue White Star's trans-Atlantic service, she gets requisitioned at some point as a troop ship or a hospital ship, and despite a close call or two with a U-boat, she probably survives the war to resume her normal passenger service, which she spends the 1920s & early 1930s, doing, possibly undergoing some refits to modernize her passenger accomodations and other alterations to reflect changing market conditions beyond normal structural & mechanical repairs.
However, the Depression hits the trans-Atlantic passenger service hard, and in 1934, White Star is compelled to merge with its rival Cunard by the British Government. By this time, Titanic is getting old and worn out, and as part of cuts to the combined fleet, she and many older ships (as well as most of the former White Star fleet) are soon taken out of service and slated for disposal in favor of newer, faster, more economical ones. As such old, large, and expensive to operate liners are unlikey to find buyers in the mid-1930s, Titanic is sold for scrap around 1936 or so, although some of the woodwork and furniture from the first class accomodations may be bought and used by a hotel or inn somewhere.