
History:
The British set foot in India around beginning of 17th century long after the Dutch and Portuguese. The main intention at that was for trade. East India Company (EIC) in 1612 approached mughal emperor Jahangir for trading rights in Surat and other areas. Initially EIC faced intense competition from the Dutch and Portuguese but was eventually able to supersede them. EIC established trading posts in Surat (1619), Madras (1639), Bombay (1668), and Calcutta (1690). Around the mid of 18th century is when EIC slowly expanded their control over weakening Mughal and started gaining territories.
British mostly took advantage of weakening Mughals and conflicts between various kingdoms and princely states to quickly gain territories in India piece by piece. In 1799 EIC forces defeated and killed Tipu sultan, King of Mysore. And in 1803, following Second Anglo-Maratha war resulted in British capture of Delhi and control of large parts of India. EIC continued ruling until the Indian Rebellion of 1857 after which the administration of India was shifted from EIC to direct British Crown under Victoria.
In 1700 India's economy had a 24.4% share of world income, the largest in the world where as in 1913 India's economy had a 5.4% share of world income under the British Empire. The british ruled India until 1947 until the effects of World War II and Mahatma Gandhi's popular Quit India Movement forced the british to grant independence to India.
Coinage:
East India Company (EIC) first produced hammered (dump) copper coins of 1 and 2 Paisa Denominations Starting 1672 from Bombay Presidency. Some of coins of this period were also struck in Tin. Generally these coins are very expensive and rare to find (especially in good grades).

(Bombay Presidency Pice 1672)
Bengal and Madras presidency too started issuing their coinage during beginning of 18th Century. Interesting to note here that generally the higher denomination coinage (such as a rupee) was struck in the name of the current Mughal emperor that had almost no power over that territory. It is often confusing thus to differentiate coinage of British, Maratha and mughal issues especially for coins which did not have dates in them.
In 1791 the first machine struck coins were produced for Bombay Presidency in 1/2, 1 Pice and 2 Pice denominations. Beginning 1803 1, 5, 10 and 20 cash coins were issued from Madras presidency.

(E.I.C. Pice of 1791 - From HA.com)
In 1835, EIC combined the presidency coins to produce Unified coinage in the name of William IV for Silver coins, whereas the lower denomination copper coins were continued to be minted with "EAST INDIA COMPANY" name. In 1840 coins were minted with Victoria's portrait. After end of ruling of EIC, the first coins of direct British ruling were minted in 1862. Imperial issues bear obverse portraits of Queen Victoria (dated 1862–1901), Edward VII (dated 1903–1910), George V (dated 1911–1936), and George VI (dated 1938–1947). No British India coins were issued during the brief reign of Edward VIII.

(1862 Rupee - Victoria - © Joseph Kunnappally)
In general coins of British India are highly desirable specially in high grades. There are many die variations, mules of this series exists that are almost impossible to get in decent prices.
https://en.numista.com/catalogue/inde_britannique-1.html




