British rule in India advanced in stages - in 1757, the East India Company forces won the Battle of Plassey, in 1767, when the company was granted the right to collect revenue in Bengal and Bihar or 1818, when the EIC defeated the Maratha forces. East India Company controlled/ruled India up to 1858, after which India was governed directly by the British Crown.
Up to 1835, different presidencies of British India - Calcutta, Madras and Bombay, used to issue their own coins - in the name of Mughal emperor Shah Jahan. From 1835-1947, India followed a uniform coinage system, with coins featuring the British ruler (William IIII, followed by Victoria, Edward VII, George V and George VI).
These coins were originally minted in three mints - Calcutta, Bombay and Madras - with their own mint marks. The Madras mint was shut down in 1869, while a new mint was set up in Lahore in the 1940s. Read more about mints here