I've got the lyrics, but not the author. My father used to sing it and I have no idea where he got it from. However, I heard a rendition of it recently on our National radio station, CBC (Canada)--I believe it was Tommy Makem singing it. Of course that doesn't mean he wrote it.

RED VELVET STEERING WHEEL COVER DRIVER
As recorded by Jeremy Taylor on “Piece of Ground” (2013)

I remember the time I was driving along
On the Liverpool Highway, singing a song,
And the day was as good as the road it was long.
I was on my way,
And then up ahead a black speck appeared,
And as I drew nearer, ‘twas just what I feared,
Doing 25 miles an hour in top gear
In the outside lane:
Oh, dear me! Glory be!
One of them, I could see:
It was a red velvet steering-wheel cover driver.

He and his wife, they sat side by side.
In their safety belts they were safely tied,
And together they drove through the green countryside,
Which they did not see.
They had two dingle-dangles which hung by a string
From the mirror, and on the back seat was a thing
Which wobbled its head up and down with a sing-
Ular stare at me,
Up and down, side to side.
Keep your eyes open wide
For the red velvet steering-wheel cover driver.

He's got three china ducks on his sitting-room wall,
And a bowl of pink plastic flowers in the hall,
And a souvenir ashtray he bought in Porthcawl
On his holiday;
And when he gets home, he'll go straight to his chair,
Switch on the telly and all night he'll stare
At the girls and the gunfights and wish he was there,
But he's here to stay.
When he regrets what he didn't see,
Then he’ll switch to ITV,
‘Cause he's a red velvet steering-wheel cover driver.

Each Monday morning he takes his basket
To town; he goes to the supermarket.
He walks up and down, thinks, “Oh, what a task! It
'S so hard to choose,
‘Cause there’s apple sponge pudding all topped with Tastee-Freez,
All of those fifty-seven varieties,
Fresh today,
Hermetically sealed, untouched by human hands, so easy, just heat the tin.”
Think of those gifts you might win
If you're a red velvet steering-wheel cover driver.

Well, it's Saturday night; it’s the night for a fling,
So let’s on with our glad rags, ring-a-ding-ding;
Go down to the hall for a good game of bing-
O and cups of tea.
Then we’re back home to bed switch out the light,
Hug our hot water bottle, sweet dreams and sleep tight,
And God keep us from things that go "bump" in the night,
And especially
All those dreams of the day
When our own children will say:
"My dad's a red velvet steering-wheel cover driver."

Corrections made based on later comments. --Mudelf, 02-Oct-18.