I was reading Wikipedia's article on Modern history of East Asian martial arts and specifically under the section "Western interest", and it mentions there was very little knowledge of Asian martial arts prior to 1959 in the West until the publication of the James Bond novel "Goldfinger". This is very likely to be true. Movies and t.v. shows such as the Green Hornet which starred Bruce Lee as Kato, the various Bruce Lee movies, the James Bond movies, Kung Fu starring James Caradine, the Pink Panther movies, and perhaps some episodes of the Batman t.v. series all appeared in the 1960s and 1970s. But there were already many Asians living in America prior to the 1960s. Moreover, the Philippines was under American colonial rule for about 50 years, and prior to that Spanish rule for over 300 years. Many other European countries had colonies in Asia also, and have had some level of Asian immigrants. Hawaii became a territory of the U.S. in 1898, and the coming decades would see the arrival of various Asians including Filipinos to Hawaii, and yet Filipino and Asian martial arts in general was not well known in the U.S. mainland until 1959 as mentioned earlier.
A possible counterexample would be Brazil, which is a Western country, was influenced by Japanese martial arts in particular Judo and Jujutsu probably as early as 1916. Gracie Jiu-jitsu, Vale Tudo, and Brazilian Jiu-jitsu would be a product of this, and all three helped developed the modern Mixed Martial Arts (MMA) scene that we have today. But Asian martial arts was for the most part unknown in the West. Supposedly servicemen living in Japan after World War II observed Karate, and help bring Karate to the U.S.
In general, the colonizer will impart more to its colony than the other way around. For example, the Philippines adopted modern boxing from the Americans (as well as basketball, baseball, etc), and sword and dagger techniques and training from Spain as in the attack angle numbering system, the martial system called espada y daga, and even terms such as arnis and escrima. But surely the colonizee also influences the colonizer albeit at a smaller and slower rate, and I suppose that is what happened, but it was very slow indeed.