The song may be American, but the clock was supposedly English, at least by residence if not by manufacture. Here's what
a Wikipedia article says about it:

There are two competing theories as to the origin of the Grandfather's Clock.

The most common relates to a wayfarer's inn in Piercebridge on the border of Yorkshire and County Durham called the George Hotel. The hotel was owned and operated by two brothers, the surname of both being Jenkins, and in the lobby was an upright clock of the kind now called a Grandfather clock. The clock kept perfect time until one of the brothers died, after which it lost time at an increasing rate, despite the best efforts of the hotel staff and local clockmakers to repair it. When the other brother died, the clock stopped, never to go again. It is said that in 1875 Henry Clay Work visited the hotel and wrote My Grandfather's Clock based on the stories he heard in the hotel.