The song stems from this broadside, or one like it. From Nineteenth-Century Song Sheets, American Memory.

BRITANIA'S DISGRACE AND THE TRIUMPH OF LIBERTY

COME all you heroes, where'er you be,
Who walk by land or *fail by sea, (*sail)
Come hear the words of a dying man,
For I am sure you'll remember them. Tol de rol.

In seventy-five, that fatal year,
As by our signal it did appear,
It was on the fourteenth day of May,
Our fleet set sail for America.

It was on a dark and dismal time,
Our fleet set sail for the northern clime;
The drums did beat, and the trumpets sound,
And unto Boston we were bound.

And when to Boston we did come.
We fought them by our British drum,
To drive those rebels from that place,
And fill their hearts with sore disgrace.

But to our sad and great surprise,
Like grasshoppers we saw them rise,
To fight like heroes so much in rage,
Which sorely frighten'd brave General.

Those brave and bold America's sons,
Saw death and slaughter from our guns;
"Freedom or Death!" those heroes cry-
I'm sure they're not afraid to die.

We sailed to York, as you've been told,
To the loss of many a Briton bold;
With many traitors in the throng,
False to the States where they did belong.

They told us it was a garden place,
We and our armies might with ease
Burn down their towns, and lay waste their lands,
In spite of all their Boston bands.

A garden place, boys, it was indeed,
In it grew many a bitter weed,
Which did pull down our highest hopes,
And sorely wounded our British troops.

Now I've received my mortal wound,
I bid adieu to Old England's ground;
My wife and children will mourn for me,
While I lay cold in America.

Fight on, fight on, my American boys,
Fear not Great Britain's thundering noise,
Your rights maintain from year to year,
God is on your side, You need not fear.

American Memory, 19th c. Song Sheets, Printed by Freeborn Watson, no date.

Many patriotic ballads came out in the 19th c. in connection with the Centennial celebrations in 1876 and the Exposition in Philadelphia.
In the period of the Revolution, there were similar songs, extolling either the Americans or the British troops. Whether "Dying British Sergeant" and its variants under several names came from Revolutionary War days or later, I don't know.

One 18th c. song, of the ilk, is "A Song, Composed by the British Soldiers, after the fight at Bunker Hill, June 17, 1775."

There are several buried in American Memory under the horrendously long titles popular at the time.