These lines have been mangled so that is is not easy to make out the sense. This version of ‘Scotland the Brave’ is often (mistakenly) credited to singer John McDermott, but the words come originally from the poem ‘Scottish National Melody’ by James Hyslop (1798–1827). The earliest printing of Hyslop’s poem that I can find is from 1821, where the verse in the question appears as follows:
Bold hearts, and nodding plumes,
Dance o’er their bloody tombs—
Shining in blood is the red tartan’s wave—
Dire is the horseman’s wheel,
Shivering the ranks of steel—
Still victor in battle-field, Scotland the Brave!
Anonymous (James Hyslop). ‘Scottish National Melody’. In The Edinburgh Magazine and Literary Miscellany, April 1821, p. 361.
There is another publication in 1836, where the wording is slightly different:
Proud heart, and nodding plume,
Dance o’er the warrior’s tomb,
Dyed with blood is the red tartan’s wave,
Dire is the horseman’s wheel,
Shiv’ring the ranks of steel,
Victor in battle, is Scotland the Brave!
James Hyslop. ‘Scottish National Melody’. In Glasgow University Album (1836), p. 3. Glasgow: John Smith.
In both these versions, ‘shivering’ means ‘shattering, breaking’. That is, the cavalry manoeuvre (‘the horseman’s wheel’) is breaking up the infantry formation (‘the ranks of steel’, referring to the pikes or bayonets), killing and scattering the soldiers. This explains why the maneouvre is ‘dire’ (dreadful, horrible, terrible), explains why Scotland won the battle, and is a more dramatic image than the modern version, in which ‘shivering’ seems to mean only ‘trembling with fear’.
These versions are improvements over the modern version in various ways. ‘Dance’ is better than ‘wave’ since it can apply to the ‘heart’ as well as the ‘plume’ (and it avoids a repetition of ‘wave’). ‘Red tartan’ makes sense since it is ‘dyed with blood’, whereas ‘green tartan’ has the unfortunate implication that the foes are Vulcans rather than Englishmen. ‘Deepeyed’ seems to be a mistake or misprint for ‘deep-dyed’†. In summary, I have no idea what McDermott could have been thinking in using the version quoted in the question instead of either of the versions written by Hyslop. I can only guess that McDermott never read Hyslop, and got the poem through an oral tradition instead.
† ‘Deep-dyed’ may be necessary for modern English speakers, but Hyslop probably intended ‘dyed’ to be pronounced with two syllables, ‘dye-ed’, so that it fits the rhythm.
The 1836 printing of the poem came with this note:
The Editors, however, deem it right here to advert to the highly beautiful national verses opening the volume, as being invested with a peculiar interest, from the circumstances of their being posthumous production of James Hislop, author of the well-known and much admired stanzas entitled “The Cameronian’s Dream,” composed when the writer was a shepherd boy in his native country of Ayr. This highly promising individual, when on his way a few years ago to a foreign shore, was shipwrecked and case on an island of savages, from whom he met a barbarous, and revolting—a nameless end.*
Anon (1836). Preface. In Glasgow University Album, pp. vii–viii. Glasgow: John Smith.
* This seems very doubtful. James Hadden wrote in the Dictionary of National Biography that Hyslop “died of fever off the Cape Verd Islands” during a voyage to South Africa.