What is known for sure is that the British Invasion began in America in February 1964, when the Ed Sullivan Show booked the Beatles for an unprecedented three appearences for the sum of $10,000. The Beatles, with their moptop hair, cheeky attitudes, and Edwardian suits became overnight sensations of the sort that doesn't happen anymore in the age of fragmented media. When the Beatles appeared, everyone saw them, from teenagers to their parents to their grandparents. And their "yeah yeah yeahs" and alien accents, and manic exhuberance, and tuneful melodicism, and electric guitars changed the musical landscape forever, big time. They made the cover of magazines, their albums shot to number one, at one point in 1964 the Beatles had all five #1-#5 singles on the chart simultaneously, a feat nobody has come close to approaching ever since.
All of this seemed very sudden in America, which was subsequently inundated. Aside from six Beatles singles to reach #1 in 1964, other #1 artists that year included The Animals, Manfred Mann, and Peter & Gordon; in 1965, The Beatles, The Rolling Stones, The Dave Clark Five, Herman's Hermits, Wayne Fontana & The Mindbenders, and Freddie & the Dreamers all reached #1 as well. Dozens of other British acts made the top-10, top-40, and top-100; the British dominance of the American charts was rivalled only by The Beach Boys and the Motown soul groups. It wasn't until mid 1965, when the first American folk-rock began to chart (itself a hybrid of Beatles-style electric rock grafted to traditional American folk music), that American rock music regained serious popularity again. British bands continued to swamp America however, with new arrivals continuing through the 1960's; the original "British Invasion" is usually considered to have lasted from 1964-1968, until the dawn of the progressive-rock era.
While Americans remember the British Invasion as appearing out of nowhere, in the cold winter following the assassination of President Kennedy, the British Invasion (which obviously wasn't thought of as an 'invasion' in England) had been developing since the late 1950's, and had reached its maturity in 1962 in England, the year the Beatles first charted in the U.K. with "Love Me Do" (a #23 hit in England in 1962)
American rock 'n' roll had made the trip across the Atlantic in the other direction, to England, in the late 50's. Acts like Elvis Presley, Chuck Berry, and Little Richard not only were popular in England, they remained so even after their popularity had waned (or been purposely sabotaged) in the U.S. In addition, many of the U.S. electric blues performers like Muddy Waters, Elmore James, and Howlin' Wolf found audiences in England that they lacked at home; many of the still living electric bluesmen of the 50's found their talents in demand in England in the early 60's. Particularly in the port cities of England, which always had a healthy influx of imported blues, rock, and pop records from America, young people were exposed to a grittier, more earthy rock experience than teens in America were during the chilly years of 1959-1963.