I'm not sure what were Neville Chamberlain's opinions on India & Indian Independence but one possible POD would be that Neville Chamberlain is still able to keep Britain out of the war, either because Hitler is slower at eating up Sudetenland or because Chamberlain is keep the course os rapprochement with Hitler, despite the latter's successes against the rest of Europe. Without expending manpower and resources on World War II, which would just become another European war (with a pacific war between Japan and US), the British are able to smarten up on the Independence movement and decide to raise local Indian leaders who will thwart Independence efforts. So from 1938-1948, they basically try to (ineffectively) suppress the Indian independence movement, while trying to get their own agents involved into the movement to render it inert.
This will blow up in their face as soon as even the local Indian leaders condemn the next "accidental" massacre by the British Raj Government. The local leaders would then demand independence for their respectively region and come up with an agreement to divide up India in an orderly fashion over a period of 10 years from 1948-1958. One of the leaders would be the Nizam of Hyderabad Mir Osman Ali Khan, who would rule over the state of Hyderabad Deccan (this state also controls the eastern coast from Vizagapatnam to Nellore, but had to give up the cities of Ahmednagar and Aurangabad to the nation of Maharashtra.
Other states would include -
-Kingdom of Mysore, ruled by Maharaja Jayachamarajendra Wodeyar.
-Maharashtra nominally ruled by a federation of Maratha Chhatrapatis but actual power is held by Independence leaders like Laxman Vasudev Paranjpe, Vinayak Damodar Savarkar, Bhimrao Ambedkar, amongst others (Savarkar leads the conservative factions and Ambedkar leads the reformist faction while Paranjpe tries to take the centrist path between them).
-Bengal, a dictatorship under Subhashchandra Bose and his cadre of bengali fascists (also includes the North-eastern states and parts of Orissa).
-Travancore, a Malayali (Keralite) socialist nation led by Koyapalli Kelappan Nair.
-Madras Presidency, a Tamil dominion of Britain led by the Justice Party.
-Sindh and Kutch, a Gujarati-Sindhi nation led by the triumvirate of Vallabhai Patel, Mohammed Ali Jinnah and Mohandas Gandhi (Patel leads the hindu faction and Jinnah leads the muslim faction while Gandhi acts as the glue that helps them stick together.
-Goa, a Portuguese colony.
-Punjab, a multi-religious Socialist nation lead by revolutionaries like Bhagat Singh, Sushila Trehan, Inayatullah Khan Mashriqi and Shambhu Datt Sharma.
-Baluchistan, a tribal federation.
-Kashmir and Jammu, a princely state lead by Maharaja Hari Singh who has to face a rebellion lead by Mohammed Abdullah Sheikh.
-Hindustan, led by Jawaharlal Nehru and is the largest nation which stretches from Delhi in the north to Indore in the south and from Jodhpur in the West to Patna in the East.
-Burma, Sikkim, Nepal, Bhutan and Sri Lanka which are just as independence OTL
Now if only someone could draw a map....