There are no easy answers here I'm afraid. Always the dilemma when you are dealing with a surname like White, people not marrying, changing their christian name/s.
Since what I found seemed to be the only option we could see, if it was me, I'd be ordering the birth certificate of John George White Sept.1893 Fulham to see if he had a birthday of 30th May.
For instance, even if it did turn out to be the right John George - with a father not Charles Edward but Edmund/Edmond Joseph, things aren't going to get any easier for you. The latter states he was bc.1856 Islington or Hammersmith but no-one to fit these details either - a number of trees on ancestry state he's the son of a Charles White and Sarah Walkley BUT their son was born and baptised as Edmund James, born 27/1/1858 St Martin in the Fields, bp.14/2/1858, so not sure Edm Joseph is even this man! See what I mean?
Annette