The Road to Clady is a version of the Scottish song The Soor Mulk Cairt. I've tried quite a few variations of words and spellings, but the original doesn't seem to be in the DT. This is how the McCalmans sing it, pretty close to the original:

THE SOOR MULK CAIRT
(Thomas Johnstone)

Chorus:
Wi' her cheeks sae red and rosie, and e'en sae bonnie blue
Dancin' and glancin' she pierced me through and through
She fairly won ma fancy, and stole awa' ma hert
Drivin' into Glesga on a soor milk cairt

Oh I am a country chappie and I'm serving at Polnoon
On a fairm near to Eaglesham, that fine auld-fashioned toon
Whaur, wi' the milk each mornin', a little after three
We tak the road richt merrily, my auld black horse and me

I raised her up beside me and we soon got on the crack
I slipped my airm aroond her waist as by my side she sat
I telt the auld, auld story as the woods around me rang
Wi' the singing o' the mavis and the blackbird's cheery sang

I've heard o' lads and lasses making love in shady bowers
And hoo they woo'd and won amang the roses and the flowers
But I' ne'er forget the morning wee Cupid threw his dart
And made me pop the question in the soor milk cairt

Since the lassie has consented next term time cams roon
I mean tae buy a harness plaid and a bonnie silken goon
We're settlin' tae get marret just aboot next August fair
When aw oor auld acquaintances I hope to see them there

The lass had never had a hurl in a carriage aw her days
Sae when that I proposed tae hae a coach and pair o' greys
Na, na, quo' she, The siller's scarce, ye ken we canna spare't
And I'd rather hae the jooglin' o' the soor milk cairt

Here's some background to the song:

[1959:] I have much pleasure in informing you that my father Thomas Johnstone, is the author of The Soor Mulk Cairt. He wrote it in the 1880s for J.C. Macdonald, the popular comedian. My father was born in 1850 [...]. The Soor Mulk Cairt was sung all around the countryside, especially by farmhands, and so often that it's not surprising that some of them claim authorship. I've had to check this publicly with the Glasgow 'Evening Times' and the B.B.C. The slow walking pace milk cart was the type used by my father to travel to Eaglesham on Sunday mornings. [...] I enclose an old newspaper cutting to prove my claim. (Letter by Mary M. Johnstone, Weekly Scotsman, August 6)

[1974:] The words of this song were written by Tom Johnstone (1850-1911), otherwise known as 'The Calton Barber Poet' or 'Figaro'. Johnstone established a barber's shop in Abercrombie Street, Calton, Glasgow, during the early 1870s, and about the same period he also became well-known as a songwriter. His shop became a mecca for music-hall artistes, who could rely on Tom to speedily manufacture a song on any subject which took their fancy. Tom had a house in Eaglesham, Renfrewshire, where he spent his week-ends. On Monday mornings he was in the habit of obtaining a lift into Glasgow on the milk-cart which left Eaglesham at 3 o'clock. The song was based on a real romance between Dan Steel, the cart driver, and Maggie Watt. The words were originally published in a Glasgow weekly periodical 'The Professional and Authors' Journal' which printed many music-hall songs. The tune is an adaptation of the Irish The Garden where the praties grow. (Notes The Clutha, 'Scots Ballads')

[1990:] The sour milk cart was drawn at a reasonable pace by one horse, of a steady disposition and 'a certain age'. Giving the driver the chance of clicking with a lass. The sweet milk cart came at a spanking speed, pulled by two fiery young horses. The sweet milk needed to be in town early, to get into the morning teacups of the well off. The sour milk was used for baking. (Ewan McVicar, One Singer One Song 96 - Tune Jamie Raeburn)