They're moving Father's grave to build a sewer is a classic music hall song. The origin seems to be unknown but was revived by the Clancy Brothers. It was a favourite, well before the Brothers' time, of Frank Muir, who sang it many times on BBC Radio 4. It is usually sung in a straight key of C major, but can be varied.
Lyrics
Lyrics vary, but Muir's version is thus:
Recorded versions
A variant of the song, called "Grandpa's Grave", was recorded by the comedian Peter Sellers and included on his 1960 LP with Sophia Loren, Peter & Sophia, as well as on the B-side of the duo's hit single "Goodness Gracious Me".
A version of the song was also recorded by Oscar Brand, on his album Bawdy Songs Goes to College recorded in 1955, under the title "Father's Grave".