So far as I know, there were no political groups in British India openly opposing independence. There were indeed attempts to derail and sabotage the process of independence, but these came from rulers of the princely states.

At the time of independence, about a third of the subcontinent was composed of some ~600 princely states, ruled by rajas, nawabs, nizams. A hundred million people lived in these little kingdoms. Each was recognized as an independent state by the British, each had its own treaty with them.

These rulers were scared of independence, for good reason. They had survived by cozening up to the British, some (such as the Nizam of Hyderabad) had even lent their armies to the British to subdue other Indians. They were worried what would happen when the British left and they found themselves situated next door to an Indian government. Furthermore, these princely states were autocracies, often ruling rebellious subjects with an iron hand, and they were concerned how their own populations would react when democracy became established next door. So there was a period of scheming and negotiations to stall Indian independence, or create hurdles which could be used to blackmail Indian leaders into giving them good terms.

They raised their concerns before the British Crown, and before Lord Mountbatten was sent as Viceroy to India, he was summoned for an audience with George VI, to address the princely concerns. Mountbatten was unsympathetic – he had previously described these rulers as absolutist autocrats and degenerates – and he told George VI that his policy would be to tell them to make their peace with the governments of India or Pakistan when they were created.

When Mountbatten arrived in India, he was met by a delegation representing these rulers, chief among them the Nizam of Hyderabad, the Nawab of Bhopal, and the Raja of Travancore. They demanded that Britain continue to regard them as independent states and renew their treaties, even after the British left India. They asked for dominion status, to put them on parity with India and Pakistan, which would be created after independence. Mountbatten refused.

These rulers panicked. They realized they were weak, there was much talk of uniting the princely states to form a union that would be comparable to India and Pakistan. It went nowhere because none were willing to give up their own authority. But then they found sympathy for their cause with Sir Conrad Corfield, private secretary to Mountbatten, and also Political Advisor to Lord Wavell, the Crown’s representative to the princely states. Corfield was a staunch royalist, and favored their cause.

Corfield started by destroying all records the British kept on the princely states and their rulers. These records contained a lot of unsavory information about them, and he feared it could be used to blackmail the states at a later time. He presented their cause to Mountbatten and argued constantly to give recognition to the princely states, but Mountbatten wouldn’t budge. Finally, Corfield bypassed him and found an ally in Lord Listowel, who was Secretary of State for India in British Parliament.

With Listowel’s help, he introduced a new clause in the Indian independence bill, which stated that Britain would continue to recognize the independence of the princely states and maintain diplomatic relations with them until the morning of independence day. This meant that on the day Indians woke up to find themselves independent, they would also find ~600 de facto fully independent princely states scattered throughout the country, each free to negotiate diplomatic relations with Britain or any other country. Bhopal, Hyderabad and Travancore immediately declared their intention of remaining sovereign states after Indian independence, and went so far as to prematurely appoint trade representatives to foreign nations.

When the Indian National Congress found out, they were furious and many accusations were hurled at Corfield. Mountbatten was unhappy too, but the change had come from London and there wasn’t much he could do, he already had his hands full. So the INC sent two of its representatives - Sardar Vallabhbhai Patel and VP Menon - to make rounds of the princely states and get them to sign an instrument of accession, which would join them to India under favorable terms.

They found strong support in Rajasthan. The Raja of Bikaner in particular readily signed and then joined the delegation and went around persuading his fellow rajas to sign. In the end, almost all the princely states in the region that was to become India had signed, with the exception of Hyderabad, Bhopal, Travancore, Junagarh and Jodhpur.

Eventually, Travancore came around because of internal rebellions, since there was strong support for joining India among the populace. Bhopal was in a precarious position, having a Muslim Nawab ruling over a majority Hindu population, so he realized that things would not go well for him after independence, and he signed too. Jodhpur’s raja changed his mind under pressure from his own ministers. Kashmir’s raja held out as long as he could until he could, but eventually signed. So on independence day, the matter had mostly been sorted out and almost all the states were on board.

Junagarh was taken when the Indian military crossed it border and camped just inside. The Nawab panicked and fled to Pakistan. There was a plebiscite among the population, over 99% voted to join India, so that was that. Hyderabad was the only state that held out for long. It was huge. But the populace was unhappy. The Nizams had been slowly Islamizing it for the past several decades, and the majority Hindu population didn’t like that. There was a peasant rebellion back in 1945, which was brutally put down by the Nizam’s Razakars. It didn’t help that much of the Nizam’s army wasn’t even Indian, it consisted of Arabs and Pashtuns. People felt that they were still under foreign rule.

India could not tolerate a huge independent state like Hyderabad right in the middle of the country, especially since Hyderabad had made many previous overtures to Pakistan. There was a period of negotiation, in which India made several offers, including an offer to let the Nizam continue to rule Hyderabad, but to relinquish control of foreign affairs to India. The Nizam refused. So in Sept 1948, India invaded Hyderabad. Four days later, most of the Razakars were dead or had fled, the Indian army was camped on the Nizam’s doorstep, and he surrendered. Hyderabad was annexed.

So while there was no overt opposition to the independence movement in India, there were attempts to sabotage the process from the rulers of some princely states, primarily Hyderabad, Bhopal and Travancore. They didn’t affect the timetable of independence, since neither George VI of Britain nor his Viceroy Mountbatten wanted anything to do with the princely states after they left. They did cause some consternation in the INC, but the INC worked to create private deals with a number of sympathetic kingdoms, and used them to isolate the few holdouts. All it seems to have done is to make heroes out of Sardar Patel and Menon.

A good source for this stuff is Leonard Mosley’s book Last Days of the British Raj.