In the UK, there is a democratically elected prime minister but the Monarch is the Head of State, although the latter is not (directly) involved in government. Likewise, the UK could have done the same in India, establishing an Indian-run Government but with the British Monarch as its head. Why did the UK not choose this option when granting independence to India?
That's an interesting question. When India first gained independence from the United Kingdom on 15 August 1947 it was as The Dominion of India, with King George VI as king and Head of State.
India became a sovereign democratic republic when the Constitution of India came into force on 26 January 1950. This repealed the Indian Independence Act, and removed the king as head of state.
The Constitution of India was drafted by the Constituent Assembly of India, which, in turn, was elected by elected members of the provincial assemblies.
So it wasn't the UK that allowed India to be a republic, it was the elected representatives of the Indian people themselves who chose to be a democratic republic.
At the 1926 Imperial Conference, Britain and its dominions had agreed they were:
"... equal in status, in no way subordinate one to another in any aspect of their domestic or external affairs, though united by common allegiance to the Crown".
Given that, the UK could hardly insist that any of its dominions should always remain a dominion, or complain when one chose to become a republic.
Because that's what independent means. There were only three options:
- keep India as a dominion, with the monarch as head of state;
- give India independence and forbid them from having the monarch as head of state;
- give India independence and let them choose whether or not the monarch is head of state.
The second is a possibility, since the monarch could simply refuse to act as head of state of the newly independent country, even if India wanted to be a constitutional monarchy. But there's no practical way that Britain could compel an independent country to adopt the monarch as their head of state, and nobody would call the country "independent" if Britain did that.
Because each dominion decided for itself when it became independent. Australia and Canada retained the queen/king as the head of state. But Ireland and South Africa elected to become republics, when they became independent (after the other two).
Compared to the others, India was the latest to become independent, and when she did so, she opted for a "republican" form of government, like the second, later pair, of countries in the previous paragraph. If Britain had managed to dictate otherwise, India's "independence" would not have been real.