To what extent did political authorities influence the course of the protestant reformation in the sixteenth century

There was significant influence by political authorities in the course of the Protestant Reformation.

During the time that furor erupted over Martin Luther's 95 Theses and other writings, the Holy Roman Emperor, Maximilian I, died (in January, 1519).  This meant there would be an election for a new Holy Roman Emperor.  This was not an election like we think of today.  There were seven high ranking persons in the German territories (princes and church authorities) who were "electors" who had votes in the process.  The prince of Luther's territory, Frederick the Wise of Saxony, was one of those electors.  So the pope was tending to be less demanding toward Frederick than he might have been otherwise, because he was trying to influence Frederick's vote in the choosing of a new Holy Roman Emperor.  Pope Leo X did not want Charles V of Spain to get that title, because Charles already held much power. 

In the end, however, Charles did become Holy Roman Emperor.  He, too, played a role in influencing the course of the Reformation by calling Luther before the Diet of Worms in 1521, and ultimately going to war against the Lutheran princes in the Smalcald Wars (1546-57).

Princes in the German territories who adopted Lutheran beliefs became instrumental in promoting and protecting the Protestant faith.  So also did others later, like King Gustavus Adolphus of Sweden, who led his nation's armies to rescue German Lutherans in the 30 Years War.

Throughout the Reformation years, there was much interplay between politics and the matters of religion that were being fought over.

The answer is protestant reformation.