I found the lyrics here: http://sallyodgers.50megs.com/lalsdrum.htm. I'm guessing that it's not a traditional song, but I have no clue as to the origins of the song.
I did find a script for Episode 1 of Harlots that makes mention of an excerpt of the song:

http://www.springfieldspringfield.co.uk/view_episode_scripts.php?tv-show=harlots-2017&episode=s01e01

-Joe-

FOLLOWING THE DRUM
(songwriter???)

Oh, once I slept in a feather bed
With woolen blankets warm
Now I'm glad to lay my head
On a cloak that's old and torn
Oh, sing with me a merry catch
As summer days will come
My love he is a soldier boy
And so I'm following the drum.

With a patter patter tat
And a rattle and a scrap
And the beat of the marching song
I'm ragged and worn
And my hose are torn-
Yet I am following along.

Oh once I ate from a silver plate
With sauces rich and rare
Now I'm glad if I'm not too late
To find some bread to share
Oh sup with me on a scraggy fowl
And dream of sugar plums
My love he is a soldier boy
And so I'm following the drum.

With a patter patter tat
And a rattle and a scrap
And the beat of the marching song
I'm ragged and worn
And my hose are torn-
Yet I am following along.

Oh once I wore a silken shawl
And a bonnet of ribbon trim
Now I'm glad if I'm clad at all
'Twixt my ankles and my chin.
Oh sew with me in a sturdy stitch
And do not prick your thumb
My love he is a soldier boy
And so I'm following the drum.

With a patter patter tat
And a rattle and a scrap
And the beat of the marching song
I'm lame and tired before my time
But still I follow him along.

Oh, I'm lame and tired but I still smile-
And still I follow him along.