Since the beginning of The Church of Jesus Christ of Latter-day Saints, when a prophet dies, the most senior apostle steps up to fill the vacancy. But, the process is still a bit more complicated than that.
With the death of President Thomas S. Monson the top leadership structure of the church is transformed, temporarily. Monson and his two counselors made up The First Presidency. When the president of the church dies, The First Presidency dissolves into The Quorum of the Twelve Apostles, the second highest governing body in the church.
As of right now, the apostles, rather than The First Presidency, lead the church. And Russell M. Nelson, the longest serving apostle, leads that quorum. Which means he’s technically in charge, but he’s not yet the president.
Church members need to wait on the official announcement which comes after all the apostles meet together. Then the new president and prophet — those titles are used interchangeably — will choose two new counselors from the apostles and The First Presidency will again take the helm.
The wait has less to do with who will be the prophet and more to do with who he, in this case Nelson, choosing to be at his right and left hand side.
The last time this change occurred it took about a week.