As far as I can tell, this was a rhyme going round British schools starting in the mid/late 1950's. I can't find an original source. Including the original poster and my conversation with my mother and aunts (as mentioned in my comment), I can find four references to this rhyme each with different words. This would be understandable for a rhyme being passed by word of mouth between school children. I can't find any trace of a second verse.
Three of the sources (my mother and the two I've since found on the internet) agree that the rhyme was sung to the tune of Hernando's Hideaway (although, to be fair, I found the other two by searching for "Quatermass" "Hernando's Hideaway" which will bias the results). The timing fits. According to Wikipedia, Hernando's Hideaway was popular in the UK charts in late 1955 (one recording reaching number 11 in October and another reaching number 1 in November). The TV series for Quatermass II, which has a plot that fits the lyrics, was broadcast in October and November 1955. Given this was going round schools, I think the TV series/UK charts match is a better fit than the film versions of each (Pajama Game in August 1957 and Quatermass 2 in May 1957). Also, Philip Hindley's comment (linked below) links the emergence of the rhyme to kids watching the TV series.
Overall, the four versions are:
There was a man from outer space,
He had a scar upon his face,
The scar was caused by poisonous gas,
His name, Professor Quatermass.
As noted by the poster, the two of the line aren't a good fit to the plot. It wasn't a man, let alone Quatermass, who came from outer space but an alien on a meteorite and Quatermass was never infected.
The version my mother and aunts recall:
There is a thing from outer space.
It leaves a mark upon your face.
It heaves and sighs its poisonous gas.
It's caught professor Quatermass, alas!
Notably they initially remembered the last line as starting "It's called" which doesn't make sense. "He's called" would give the same plot mismatch as the original poster. "Quick call" would fit the plot but then the "alas" at the end doesn't quite fit. Misremembering "caught" as "called" is a reasonable error, notably as Hernando's Hideaway has "called" in that position. It's too long since I've seen the series for me to remember whether Quatermass is ever captured or trapped which would be needed for "it's caught" to fit the plot.
From a posting by Roger Thornhill on filmdope.com:
There was a man from outer space
Who had a scar upon his face
This scar was caused by burning gas
They call him Quatermass
Roger notes, "For it to scan properly his name has to be pronounced Kway-ay-ay-ay-termass." This also has the same two plot mismatches as the original poster's version.
From a posting by Philip Hindley on missingepisodes.proboards.com:
There was a poison gas from space,
that left a mark upon your face,
its here again, its here at last,
its here the second Quatermass!
This one's interesting as it reads as more of a trailer than a telling of the story.
In general, the fit between the Quatermass rhyme and the Hernando's Hideaway lyrics is pretty good. Using the version I've heard, so I know how it's supposed to sound, I can show how to sing it by comparing it to the original lyrics. I'll put dashes between syllables/notes and use multiple dashes and spaces to line things up (oh, and memetic hazard warning: this is a bit of an earworm, every time I think of it, it gets stuck in my head for two days):
There is a thing from ou--ter space.
I know a dark se---clu-ded place.
It leaves a mark up--on your face.
A place where no one knows your face.
It heaves and sighs its poi--sonous gas.
A glass of wine a fast em-----brace.
It's caught pro-fe--ssor Qua-ter-mass, a-las!
It's called Her-nan-do's Hi--dea-way o-lé!
On line three, there's so little emphasis on the "o" in "son" in "poisonous" that it's effectively two syllables "poy-znuss".
(I originally wrote what I knew as a comment in the hope that my clues would jog someone's memory. I've since done some more searching on the internet but I still can't find an original source. I'm writing up my findings as a partial answer as per the question answering guidelines.)