A designated survivor is chosen earlier than 10 to 15 hours in advance. Even if chosen somewhat randomly, the list of people eligible to be designated survivors is created long before hand. The procedure was created back in 1947 as part of the Presidential Succession Act. Generally, however, the survivor must be eligible to serve as president and is generally a member of the president's Cabinet and will be the highest ranking member of said cabinet who survives a potential attack on U.S. politicians or a person chosen to be the designated survivor based on what politicians are planning to meet over. Jon Favreau, former speechwriter for Barack Obama, said the process was almost "entirely random", but the designated survivor can sometimes be intentionally chosen based on what the president intends to say during a speech.
tl;dr: The designated survivor is chosen much earlier than 10 to 15 hours by a process that is random or based around what the politicians plan to talk about. It is usually someone in the president's cabinet and the only time it might take 10 to 15 hours to tell an unaware person they are the designated survivor - or new POTUS as of the disaster - would be if a lot of people, including the original designated survivor, were killed & a new designated survivor had to be found based on the line of succession outlined by the Presidential Succession Act.