Jack Campin: Billy Jones and Ernest Hare were 2 of my favorite singers of comical songs. (They also did some sentimental ones.) They did not use pseudonyms (that I know of) and they mostly did not write their own songs. They often performed as a duo called The Happiness Boys.

Several songs that they performed have been posted at Mudcat:

IN THE LITTLE RED SCHOOL HOUSE
DOES THE SPEARMINT LOSE ITS FLAVOR ON THE BEDPOST OVERNIGHT
IF YOU KNEW SUSIE (LIKE I KNOW SUSIE)
SINCE HENRY FORD APOLOGIZED TO ME
THE FARMER TOOK ANOTHER LOAD AWAY
BRIDGET O'FLYNN (WHERE'VE YA BEEN?)
I WISH I WAS IN PEORIA
ALL BY YOURSELF IN THE MOONLIGHT

The Internet Archive has a big collection of their recordings, here.

I imagine Lonnie Donegan discovered them the same way a lot of musicians of his era did: by sorting through bins of old 78-rpm records at jumble sales.