Green’s Dictionary of Slang

a notional administrative measure, the British Standard Handful, supposedly used for measuring the female breast; thus by ext. the breast itself.

1977		D. Powis Signs of Crime 175: BSHs Ironically humorous play on an official sounding but mock abbreviation, in this case meaning a woman’s bosom – ‘British Standard Handfuls’; becoming a common expression.
1997		Roger’s Profanisaurus in Viz 87 Dec. n.p.: BSH abbr. British Standard Handful. The unit by which breasts are often measured.