Why did the creators of this myth begin the history of Britain by having Brutus and the nephew of King Priam build a second Troy where London now exists?
First, The Once and Future King by T. H. White, is not a myth but a novel based on a combination of earlier literate retellings (primarily Malory) of Arthurian stories that in their turn were based on oral epos. How stories came into such traditions is extremely complicated; as we have no access to the earliest oral strata of the stories, any evaluation of how and when given elements moved into the cycle are, at best, speculative.
The Trojan theme follows Virgil's Aeneid in considering the Romans descendants of the Trojans. Just as Rome was, in a sense, a second Troy, so too was London. Malory may well have been asserting the importance of England as a legitimate successor to Rome with this mythic foundation story, giving London a pedigree on par with Paris, Byzantium, Rome, and other major cities. Often foundation myths are intended as patriotic ways of emphasizing the importance of a country or city.