A loud-mouthed but ineffectual upper class fool.
Was Henry British? No, American.
This sounds very much like a phrase of British origin - possibly from the pen of P. G. Wodehouse. In fact it's an American phrase, coined by Damon Runyon. In a piece for a 1936 edition of Collier's Weekly, Runyon wrote:
"He is without doubt strictly a Hoorah Henry, and he is generally figured as nothing but a lob as far as ever doing anything useful in this world is concerned."
See also - Sloane Ranger.
See other phrases that were coined in the USA.