The issue itself recurses back to how he can actually be Viceroy.
The thing with becoming Viceroy is that it happened to men who were already working within administrative structures or who had prestigious careers. Thesiger had previously served as Governor General of Queensland and NSW, and became V&GG following a regimental posting, Isaacs was a former Cabinet Minister who had served as Ambassador to the US and Lord Chief Justice of England, Halifax was the grandson of the former Indian Secretary and had held various Ministerial jobs after he was denied the Governor-Generalship of South Africa, Willingdon had a long career in Governorships, Linlithgrow had been chairman of the Royal Commission on Agriculture in India and had such passion he became Viceroy after declining 'lesser' postings, Wavell was commander of the Indian Army, and Mountbatten was, well, Mountbatten. So for Churchill to become Viceroy, you'd need a pre-crossing the floor PoD, and for him to have been previously holding a prestigious position that got him noticed by the Monarch and recognised as a good man for such a posting. Secretary of India in a Conservative Government would be good start, but given his views on the Indian people, this may become a problem and would only prove how unsuitable he was for such a role.
If he was to become the Duke of Marlborough early in the 20th Century (during the First World War), he may be recommended for a minor Governorship (the Governorship of New Foundland strikes me as a possibility), which would give him a foot on the ladder if he proves successful. But then, if he did get to India, given what we know about his views on Indian Nationalists, things would get ugly.