If a broken toe needs to be put on a splint, how long can I wait before doing so?

Right away. If you have a broken toe, it needs to be splinted right away. Buddy taping is frequently used. Sometimes a walking boot may be helpful to further immobilize it or a surgical shoe. If it is to be casted, it may wait 2-3 days for the swelling to diminish, so it would not hand loose, however, it is to be splinted right away one way or another.
As soon as possible. Most toe fractures do not require a formal splint, but instead physicians will "buddy tape" the affected toe to a nearby toe to act as a splint. If the toe requires a more comprehensive splint and the fracture is more serious, this should be put on as soon as possible to prevent further damage and increase comfort.