According to the teachings, it's actually automatic, it happens straight away. Just like when you wake up in the morning you are automatically into your new life, you change from the death of a sleeping person into the waking of a living person. So basically it's automatic and as much it is automatic, it is taught that some people get stuck in a particular realm and therefore they don't actually get the rebirth they are supposed to get.
Now follow me on this: A very good person who when they die, they going to get a very good rebirth, based on everything good they have done, but as they are dying, they start worrying about a project they didn't finish. Buddhism does teach that often this person won't get the rebirth they deserve, they are going to get a rebirth were they are stuck - kind of like a ghost realm - hanging around the project they didn't finish. When the project gets finished, then they get the rebirth they deserve. Now to a certain extent, they deserve that middle rebirth anyhow, so it still works that when you die, you get an automatic rebirth. But the thought of getting stuck in one that we really don't want to be stuck in, that's an important little warning to keep in mind so that you work on protecting the thoughts at the moment of death. That is, you don't want to be worried about a project.