Say I wanted to make an improvised version of someone else’s copyrighted Christian worship song. How would I go about applying for the right to make an improvisation of their song, and how much in royalty fees would I likely be paying?
You would have to find the publisher of the song and send them a proposal. You can usually find the publisher by searching the ASCAP, BMI, SESAC, or Harry Fox Agency data bases. Note that if some songs have multiple composers, they may have multiple publishers and you will have to get rights from all of them.
I am not exactly sure what you mean by an improvised version, but I assume that is some kind of derivative work. Please be aware that many publishers do not license derivative works unless you are a Hollywood movie studio or something of that sort.
Most publishers will want to know what your plans are for the song and how much your budget is for the entire project. They will send you a quote. How much depends on whether you want to include the song in your wedding video, an independent documentary, or the next Star Wars film.
I am assuming that you want to change the lyrics or the music of the song.
If you just want to make a sound recording of an existing song without altering the words or the music, in the United States, most publishers are represented by the Harry Fox Agency. Just pull out your credit card and you can get a license in minutes.