Strictly speaking, the BOE doesn't produce any coins.
And neither do most monetary authorities around the world for whatever (probably historical) reason. Coinage is issued "directly" by the government (through a mint) while notes are issued by the central bank.
Cite above is why the Bank of England (and other component countries of the U.K.) issues notes while the Royal Mint produces coins. I'm sure the same setup exists for most if not all of the E.U. member states. For France,Monnaie de Paris produces coins while Banque de France (well, through the ECB) issues notes. In the U.S., it's the Federal Reserve Bank (and issued not by a central bank but one of the several branches - Boston, NY, SF) and the U.S. Mint respectively (at least two remaining mints unit recently- Philadelphia and Denver)