To answer this question I would first have to correct you that the underlining idea behind the partition of India was not lingual or ethnic nationalism but Communalism. Communalism is a concept that the basic structure of the society is defined by religion so much so that the secular(socio-economic) interests of the individual are defined by one's religion. Thus individuals belonging to different religions have different interests. This theory was propagated by the Muslim League as well as Hindu Right Wing Parties up until early 1940s. However in around late 1930s the idea that the interest of Hindus and Muslims were contradictory to each other and could not be accommodated under a single nation began to gather steam thus the Two Nation Theory proposed by M.A.Jinnah(the founder of Pakistan) stated that the interest of the Muslims could only be accommodated by the partition of India into India and Pakistan(which comprised of the Muslim majority provinces of British India namely Punjab NWFP Baluchistan Kashmir Sindh and Bengal).
Secondly the British did have a (big)role in the partition of India by using communal sentiments to weaken the National Movement lead by the Indian National Congress (which was wholly secular in its outlook), however the actual idea of Partition was not of British Origin. The demand for partition was only conceded so as to prevent the widespread communal riots that had taken place in 1946 and were spiraling into an outright Civil War.
Thus as a result we have a secular multi ethnic India and an Islamic multi ethnic Pakistan. The traditional concepts of nation states based on language, ethnicity do not apply to present day India and Pakistan.
References:
Primarily: India's Struggle for Independence, 1857-1947 by Bipan Chandra, Mridula Mukherjee, Aditya Mukherjee, Sucheta Mahjan, and K.N. Panikkar (1987).
For better understanding of India as a nation do read: Argumentative Indian, Writings on Indian Culture, History and Identity by Amartya Sen(2005)
Follow up question: if India and Pakistan become independent at the same time, why is India today much more developed than Pakistan is?
It seems that it wasn't originally just the two states which India was divided into. The Princely states seem to have been offered independence, and while the majority opted to join the Union of India some, particularly Hyderabad, tried to be independent. According to at least one source that I can find now (Wikipedia also mentions it, but that's Wikipedia) the Nizam went so far as to request help from the UN Security Council, before Hyderabad was invaded by the Indian army a year after India itself became independent
UN records back this up, talking of a dispute between India and the "state of Hyderabad":
By cablegram dated 21 August 1948,^585 Hyderabad informed the Security Council, under Article 35 (Z), that a grave dispute had arisen between Hyderabad and India, which, unless settled in accordance with international law and justice, was likely to endanger the maintenance of international peace and security. The letter stated that “Hyderabad, a State not a Member of the United Nations, accepts for the purposes of the dispute the obligations of pacific settlement provided in the Charter of the United Nations”. By subsequent communications dated 12 and 13 September, Hyderabad informed the Council of the imminence and subsequently of the occurrence of invasion.^586
EDIT: Okay, I found something showing a dispute between India and Hyderabad involving the Security council here from this:
The Case of Hyderabad Before the Security Council. Clyde Eagleton, The American Journal of International Law Vol. 44, No. 2 (Apr., 1950), pp. 277-302
There were over 500 princely states that had recognized Britain as paramount, when Britain determined to give India her independence. Collectively they covered 2/5ths of India and a population more than that of the envisaged pakistan. (93 million vs 75 million for pakistan and 225 million for rest of india)
Per the partition plan of June 1943, their status was undetermined; dissolving the british empire could leave them free. The Indian Congress leadership led by Nehru and Patel were determined that India would not be balkanized (left to chaos and fomenting revolution).
The Congress policy refuted the position that these states were sovereign entities. They appealed to Mountbatten for help. He was convinced that securing the accession was essential to a negotiated transfer of power and used all his influence to make it happen.
As a relative of the King, a personal friend to many princes and the continuing head of state of post-independence India through 1948, his influence was significant. Britain would cut all ties, providing neither dominion nor commonwealth status nor arms to these states. He pointed out geographical, economic and law and order compulsions for the states to join india or Pakistan. Travancore, Hyderabad and Kashmir still retained visions of independence. Others tried to hold out or negotiate with Pakistan or India
Patel and VP Menon were responsible for much of the negotiation and the treaties. The terms of the treaties were more liberal than congress policy talk; they were not afraid of using a hardline approach when necessary either. With Mountbatten providing the carrot and Patel and VP Menon the stick, the states joined either India or Pakistan.
However, the marks of the partition plan and the dreams of independence are visible to this day in the resulting partition of kashmir and the border disputes between India, Pakistan and China.
The task of politically integrating the states began with the creation of the Indian constitution between 1948 and 1950 and political participation thereafter.
When the British left India-the states had three choices - (a)To join the Union of India (b) To join Pakistan and (c) To remain independent. Now as we know,all the Muslim majority states like Sindh joined Pakistan and Hindu Majority States joined India.The Home minister of India(in 1947) Sardar Vallabh Bhai Patel convinced many states like Jodhpur to Join the Union.States like Bengal and Punjab were divided into two parts.The Muslim majority areas like East Bengal and Western Punjab ceased to Pakistan because the Muslims feared that they will not get the same rights as Hindus.
Now there were two states who wanted complete independence-Kashmir and Hyderabad.But when Northern tribes invaded Kashmir with support from Pakistan to liberate Kashmir from the Dogra rule ,the Maharajah asked for help.India agreed to help the Maharajah on the condition that Kashmir must merge into India.Therefore Maharajah Hari Singh signed Instrument of Accession with Government of India.Indian army fought the invaders until this conflict was taken to UN. Therefore part of Kashmir which was liberated by the tribes became part of Pakistan and the rest became part of India.As for Hyderabad,the Sultan did rebel but his forces were crushed by Indian army.