While there were certainly South Asian people in the UK before 1945 - Queen Victoria having received the title of Empress of India - the period between 1945 and 1962 saw a period of sustained migration between South Asia and the UK (and also from elsewhere in the Commonwealth, such as Jamaica); people from South Asia were targeted and often sponsored by, for example, transport organisations in response to labour shortages.
And entry and residence in Britain in this period was basically unrestricted, so long as you were a citizen of the British empire and your home country didn't restrict it. 1962 saw a tightening of the legislation making it more difficult to emigrate to Britain, but the result was that, by the 1960s, there were now large South Asian communities across the United Kingdom and especially in London. In the period between 1948 and 1961 approximately 100,000 people from India and Pakistan emigrated to Britain (along with about a quarter of a million people from the West Indies).
This emigration was partly why the Beatles in 1966 could contract members of the Asian Music Circle to play Indian instruments on 'Love You To' from their album Revolver (and thus partly why an interview with Paul McCartney was talking about this in the first place).
A 1966 book called The New London Spy - something of a travel guide to London written by Hunter Davies who also wrote the official Beatles biography a couple of years later - commented on the extent to which South Asian culture had become part of London:
Indian restaurants, Indian food, Indian shops, Indian cinemas, Indian concerts, Indian plays, yoga, gurus and contemplation are now all so much part of the London scene that when a grey Bentley drew into a Swiss Cottage petrol station recently and a 6 ft. 6 in. Sikh stepped out wearing a purple turban, green raw silk coat, white jodhpurs, gold slippers and an oriental dagger with a gem-studded hilt, the Irish attendant did not bother to take more than a passing glance.
Because the BBC had long been a public broadcaster where there was an implicit part of the mission to balance catering to majority and minority interests, and because there were concerns at the time about South Asian people living in the UK being isolated and not integrating into society, there was discussions at the BBC through the early 1960s about putting on South Asian-centric programming on the BBC. Proposals to have South Asian centric programming on television was initially kyboshed because of concerns it would provoke racism against South Asian people, but eventually in July 1965 two conferences were called between BBC suits and representatives of the South Asian and Caribbean communities in London. The Caribbean community seemed to be of the opinion that they didn't need representation on BBC, as this would further isolate them from the rest of Britain; they needed better education for the British public in terms of understanding who they were and what they were about. However, because - unlike the Caribbean community - English was not a first language for many South Asian immigrants, the representatives of the Indian and Pakistani communities were in favour of specialty programming.
The BBC set up an 'Immigrants Programming Unit' in October 1965, and the result was weekly programming on the BBC's Home Service (the modern Radio 4) on Sundays from 8:10am for half an hour, and a half hour television program at 9am on a Sunday on BBC1, with a very assimilationist philosophy; the Home Service program was called Make Yourself At Home, and the programming varied from advice to new immigrants, and snippets of music from Indian and Pakistani films. It was broadcast entirely in Urdu/Hindustani, except where (on radio initially, and then on television) there were English lessons.
So, the Indian music Paul McCartney would have heard would have been Indian film music of the 1960s. My main source, Gloria Khamkar's PhD thesis from 2016, The Evolution of British Asian Radio in England: 1960 – 2004 says that there are basically no surviving snippets of this programming from the 1960s in the BBC archives; this was a period where BBC archives were regularly cleaned up and wiped or thrown out (leading to the non-existence of several episodes of Doctor Who and the like); the Monty Python's Flying Circus television shows apparently only survive because the group acquired video copies themselves. So the exact Pakistani/Indian film music being referred to is unclear, and Khamkar can't delve much more deeply into the programming based on her sources. But presumably it would have been based on the popular films of the era in India. One of the more popular songs from a Bollywood film from 1965 in India was 'Hum Jab Simat Ke', sung by Mahendra Kapoor and Asha Bhosle, and I'd suggest that the BBC played it safe with their musical choices and went for songs like 'Hum Jab Simat Ke' sung by big stars like Bhosle from popular films. Certainly the later existence of the 1990s hit 'Brimful Of Asha' by Cornershop, an British group with members of Sikh heritage, suggests that Asha Bhosle's music was well-known and liked in the South Asian community of England, and this is likely partly due to the efforts of the BBC starting in the mid 1960s.
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