Thanks to the 1948 British Nationality Act anyone from the Commonwealth, such as India or Pakistan, could migrate without visa. This caused the first wave of migrants as a response for the huge demand of doctors, nureses and workers (as in other European countries). Now, to explain the delayed effect, chain migration is the factor, the first generation brings their wifes and children, building communities, attracting the next generation to move abroad, and so on, taking place over decades. I've seen it with Turkish communities in Germany, the first generation came for the same reasons as in the UK, now they constitute 3% of the German population, but they deeply assimilated into German culture.