Saumya Malhotra, The Revolutionary's Playlist: A Lyrical Journey Through History (2017) asserts that the "tow, row, row, row, row, row" is an imitation of the sound of the drums accompanying the grenadiers on a march:
The introductory verse alludes to the great Heroes of Ancient Greece, touting the Grenadiers as past compare in their grandeur and bravery. It also introduces the repeated onomatopoeic refrain, 'With a tow, row, row, row, row, row, to the British grenadiers', which mimics the rhythm and beat of the drums in play. Further it suggests the aural imagery of soldiers plodding on, toiling away.
Rudyard Kipling, "The Drums of the Fore and Aft" (1888), a tale of two drummer boys with the British army in Afghanistan, hyphenates the initial tow-row (and in some slightly later editions the entire tow-row-row-row-row-row sequence) of the refrain:
With a tow-row, row, row, row, row,
To the British Grenadier.
which to me suggests a drum beat rather than, say, a boat being towed and rowed on a canal.
Calliope: Or, The Musical Miscellany (1788) lists five verses of "British Grenadiers," four of which have the refrain "With a tow, row, row, row, row."
Eric Partridge & Jacqueline Simpson, The Routledge Dictionary of Historical Slang (1973) includes entries for tow-row as a noun and as an adjective:
tow-row [n. 1.] A grenadier: military: ca 1780–1860. Ex 'With a tow-row, row-row, row-row, for the British Grenadiers'. cf. the adj. 2. A noise: dial. > (low) coll: from ca 1870. Reduplicated row, a disturbance. 3. As tow-row! it meant, among London crossing-sweepers of ca. 1840–80, 'Be careful, a policeman is coming!' Mayhew, Paved with Gold, 1858.
tow-row adj. Drunk (?and disorderly): C.18. Steele, 1709. On ROW, disturbance.
But as Partridge & Simpson note, these slang meanings followed, rather than preceded, the "British Grenadiers" march. Malhotra's suggestion that the sounds are onomatopoeic for the sound of military drums strikes me as reasonable. Ultimately, tow-row row-row row-row seems as successfully imitative of a marching band as ta-ra-ra boom-de-ay (another popular lyric) does.