Floats off down the river Nile
Off and down are both adverbs which modify the main verb (floats) and each other. I'm guessing the implied subject is he / the oarsman.
Off is being used as an adverb under the second definition:
So as to be removed or separated.
Other example sentences using the same definition:
He whipped off his coat = to take away
A section of the runway had been cordoned off = to be separated from
Down is being used as an adverb intensifying, or adding detail to, the other adverb, as to specify it's location (Where is he off to? Down there.). It falls under the first definition:
Towards or in a lower place or position, especially to or on the ground or another surface.
‘the sun started to go down’
‘he put his glass down’
‘he swung the axe to chop down the tree’
So floats off down the river nile can be construed as:
he [the oarsman of the Nile boat], floats off (describing how it floats = is he floating off or on the river Nile?) down (where is he going? = down the river nile) the river Nile.
Why? Because as the lyrics say in the second line:
The oar [snapped] in his hand