The text of this hymn was written by Charles Wesley in 1739 who was the leader of the Methodist movement in England in the 18th century and is best known for more than 6000 hymns he wrote.
The original opening text was a little different: "Hark! how all the welkin rings / Glory to the King of Kings" but Wesley's co-worker George Whitefield changed it to the one we use today. The modern music of the hymn was adopted from Felix Mendelssohn's cantata that he wrote in 1840 to commemorate the invention of the printing press by Johann Gutenberg (hence the name of the hymn tune - Mendelssohn).
At any rate, here is the tonal plan of this improvisation: F major, C major, A minor, D minor, G minor, B flat major, E flat major, A flat major, B flat minor, and F major (with Coda).
This straightforward plan leaves a powerful impression on the listener for several reasons - the fluency of performance, the tune is lovely and familiar and the key changes make the harmony quite colorful (especially when you change the major mode to the minor and vice versa). It works best if every verse is played on a different registration.