No, if you are studying at a London university, you don't have to be living in London to qualify for the London lodgings rate of bursary. Providing you do not live in the parental home and therefore qualify for the lodgings rate and you are studying at a London university, you will receive the London Lodgings rate.
A student lives in rented accommodation and therefore not in the parental home, in Essex and commutes to London Metropolitan University. The student will qualify for London lodgings rate of bursary.