It seems as if a shift occurred and the descriptive "Oriental" was replaced by "Asian" as the accepted term in polite society — what caused this shift?
A possible answer to this, in much more general terms and in a parallel vein to the historical account (which I do think best explains this particular case), is that de-adjectival nouns used to denote ethic groups seem to become pejorative(s).
I'm sure you can all come up with your own examples...
The New Oxford American Dictionary says on this topic:
USAGE: The term Oriental, denoting a person from the Far East, is regarded as offensive by many Asians, esp. Asian Americans. It has many associations with European imperialism in Asia. Therefore, it has an out-of-date feel and tends to be associated with a rather offensive stereotype of the people and their customs as inscrutable and exotic. Asian and more specific terms such as East Asian, Chinese, and Japanese are preferred.
I think it's simply that Oriental implies a Europe-centric view of the world, which is linked to colonialism and imperialism. Thus, it was rejected by the nations it was applied to.
I understand why the term Oriental is derogatory, but my family, who are Japanese-American, have always used the term to differentiate Japanese, Chinese, and Korean cultures, which share some common cultural threads, from other Asian cultures. For example, people from Indian are Asian, but their culture is very different from Japan, China, and Korea. Has anyone else used the term Oriental as such?
To supplement the other good answers here, I did an Ngram of Oriental, Asiatic, and Asian to provide a timeline of the various terms' use in print:
As you can see, the use of Asian increased dramatically starting in the late 1940s. One guess as to the reason is the use of Pacific-Asian Theater in popular media to define the non-European areas of WWII operations.
Also of note, former UCLA historian, Yuji Ichioka, is credited with coining the term Asian-American in the late 60s while at UC Berkeley.
Ichioka coined the term "Asian American" to frame a new self-defining political lexicon. Before that, people of Asian ancestry were generally called Oriental or Asiatic.
Of course, as you can see from the graph, Asian has been in use as long as the other terms, but developments such as the trend toward immigrant groups naming themselves by their ancestry has definitely hastened the move away from imperialistic language.
Oriental is not considered especially offensive in the UK. It is not the preferred term for a person of Asian origin or descent, but it is not one that is universally avoided.
I should add that in the UK "Asian" almost exclusively refers to South Asian (but not Indo-Chinese) origin, unless the context requires otherwise.
In any case, the term "Asian" ought not to be used as anything other than a strictly geographic descriptor, given the cultural and geographical diversity found within Asia - if you were to try to group the various countries or nations into groups that are tolerably similar to each other (a) you would have at the very minimum three groups (b) almost everyone falling under such a grouping would object to the grouping.
Perhaps it was caused by Edward Said's book Orientalism. I don't know when exactly the shift occurred as the term seems to have been out of favour for quite some time. Although, according to wikipedia, the US Senate only decided it was inappropriate for official use in 2002.