This answer only addresses the manga and Fullmetal Alchemist: Brotherhood, but it may also apply to the 2003 anime to some extent. In general, it is quite impossible to address all three canons because they diverged too far.
Amestris was founded for the explicit purpose of helping Father ascend to godhood. This is clearly stated in "The Shape of This Country" and numerous times thereafter. Furthermore, at no point is there any evidence in the anime or manga of civilian control of the military, except perhaps for Bradley himself (and he is clearly not a civilian, see for example "The Ishvalan War of Extermination" and several other episodes). We must therefore assume that Amestris is a de facto military dictatorship with Bradley at its head. Regardless of what title you want to give him, he was (publicly) in charge. However, it was not always this way, because Father founded Amestris shortly after destroying Xerxes (see "The Dwarf in the Flask" and subsequent episodes), and Bradley was far too young to have been king the entire time.
Officially, Bradley rose to his position through the ranks and achieved leadership through merit, but this is in fact a lie. He was a homunculus specifically groomed for the position, as shown in "Reunion." Before Bradley, the military collaborated directly with Father, and continued doing so when Bradley was unavailable (see the end of "Looming Shadows" and subsequent episodes). There was no actual need for Bradley to climb the ranks; Father wanted Bradley in charge of the country, and the military worked for Father.
Bradley was, in the end, a puppet of Father, created as a figurehead for the purpose of uniting Amestris and guiding it through the remaining wars necessary to set up the country-wide alchemy circle, especially the civil war in Ishval. As a human spliced with a philosopher's stone, he aged like any other human. This was critical to maintaining the charade that he was just like any other human. But it also meant that he could not be installed earlier, because he would have aged too much by the time of the Promised Day.