What does "dire is the horseman's wheel" mean?

This line comes from the Scottish patriotic song "Scotland the Brave".

Bold hearts and nodding plumes wave o'er their bloody tombs,
Deepeyed in gore is the green tartan's wave,
Shivering are the ranks of steel dire is the horseman's wheel,
Victorious in battlefield Scotland the brave.

The rest of the stanza makes sense to me in the poetic language of songs, but I don't get this line. Presumably "wheel" here is referring to "turn", but then I don't see why that would be described as "dire".

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.

I think user888379 is correct in the comments to the question.

dire, ... a. Extremely unfortunate; dreadful; terrible; as, a famine causes dire distress. [L. dirus, dreadful.] Funk and Wagnalls New Comprehensive Standard Dictionary

and

wheel, n. ...
4. A turning of a body of troops or a swinging of a line of ships, as if pivoting on an axis
Funk and Wagnalls New Comprehensive Standard Dictionary

The orders "Right Wheel" or "Left Wheel" were used when I was doing military drill (parade ground marching) to describe the gradual turning of a column of people marching 3 or 4 (or more) abreast, where the inside marchers shorten their stride and the outside marchers lengthen theirs so that the rows stay side by side and in formation as the column direction changes.

I think that a line or column of cavalry wheeling into your troops would be dreadful, and that this is likely to be the meaning.

The other answers answer the question just fine, but I wanted to add an additional image:

You're in the enemy ranks, shoulder-to-shoulder with spears and shields, feeling safe from the less organized Scottish line. Anything coming your way is likely to get shish-kabobed. Things are pretty good.

Then from the right, in between the infantry on both sides, is a line of Scottish cavalry, coming right between you and the infantry you've been worried about. What are they doing? They're turning. They're swinging your way. The entire line is swinging like a barn door and they're coming STRAIGHT AT YOU.

A horse weighing three quarters of a ton is coming at you at a full gallop. If he doesn't hit you, then the one to his right or left will. If that doesn't work, the guys on top are swinging swords. That long pike you're holding isn't going to be much good.

They said, what did they say? Stay in ranks and use your spears all together is basically what they said. Use your shield for defense. But what is that going to do?!? Oh my gosh, they're almost here--

They didn't say was going to be so loud. Someone in your left is wavering --

SMASH

The horseman are gone now. Where is James? He's supposed to be on your left. Oh, there he is --- yikes....

That's not your problem. Your problem is that the infantry line across from you is now charging, and you don't have your buddy at your side to make that nice wall of pikes that you liked so much. The other side doesn't seem to mind that their weapons are much shorter, now.

This day is not going as well as you thought.

There are probably a hundred historical errors in what's written above, but that should give an idea on an emotional level of what "dire is the horseman's wheel" means. It means by "wheeling" around toward you a relatively few brave horsemen are going to ruin your side's entire plan for the battle, and they're scary.

Bonus points if you picked up on the small Gettysburg (movie) reference in all that.