Is there a more specialized word for "expedite" for the case where the thing was not making any progress at all to begin with?
kickstart - to make something improve or become more active after it has stopped or slowed down.
Another definition that might fit OP's context better: To kick-start a process that has stopped working or progressing is to take a course of action that will quickly start it going again.
The phrase breaking a log-jam is sometimes used.
Also, if the reason for the lack of progress is bureaucracy, cutting through red tape.