In the UK, coins are produced at the Royal Mint in Llantrisant, south Wales. Coins are said to be "minted".
Banknotes are produced by a secure printer which is employed by the Bank of England to make them. Currently the Bank uses De La Rue plc, which has a LOT of experience producing printed documents to secure government specifications - it is the world's biggest company that does this and works for many governments. In Scotland and Northern Ireland, some banks issue their own banknotes and again they will use a secure printer, most likely De La Rue!