Mr National Efficiency was never known as Herbert, but by his initials (H.H.) in public and as Henry in private; 'Herbert Asquith' is a mistaken contemporary informality.
Or as wikipedia puts it, "in his younger days he was called Herbert within the family, but his second wife called him Henry. His biographer Stephen Koss entitled the first chapter of his biography "From Herbert to Henry", referring to upward social mobility and his abandonment of his Yorkshire Nonconformist roots with his second marriage. (this, presumably, is what makes him an utter bastard, my note.) However, in public, he was invariably referred to only as H. H. Asquith. "There have been few major national figures whose Christian names were less well known to the public," writes his biographer Roy Jenkins. His opponents gave him the nickname "Squiff" or "Squiffy", a derogatory reference to his fondness for drink."