The first professional baseball team in Japan was founded in late 1934 and called the "Dai Nippon Tokyo Yakyu Kurabu" (the great Japan Tokyo baseball club). According to "Taking In A Game: A History Of Baseball In Asia" by Joseph A. Reaves, the founder of the team, Matsutaro Shoriki, "outfitted his team in natty uniforms emblazoned with kanji characters instead of Western numbers". Here's a card showing Eiji Sawamura wearing this uniform:
|2006 BBM Nostalgic Baseball #067|
The team spend the 1935 season barnstorming in the US. They were fairly successful, winning 93 of 102 games against semi-pro and Pacific Coast League teams. According to Reaves - "The only minor drawbacks to the team's popularity in the States were their kanji characters and their cumbersome Japanese name. They rectified both by renaming themselves the Tokyo Giants and adopting a uniform identical to the New York Giants..."
So it would appear that the first professional team in Japan adopted an English name and used English on their uniform to become more popular during their tour of the US. I'm going to make a couple assumptions here - the Giants decided to keep the name when they returned to Japan because they had been successful with it in the US and the other teams in the Japanese Professional League that started in 1936 adopted English names with English uniforms because the Giants had done it. I don't know if that's really the case, but the other teams did have English names and did use English on their uniforms. Here's a card of Masaru Kageura of the Tigers in a uniform that dates from 1936-40:
|2006 BBM Nostalgic Baseball # 012|
Here's a couple examples - Shosei Go in a Giants...oops, Kyojin uniform circa 1940-43 and Takeshi Doigaki in a Hanshin uniform from the same time period:
|2000 BBM 20th Century Best 9 #326|
|2000 BBM 20th Century Best 9 #386|
|2000 BBM 20th Century Best 9 # 126|
Following the war, the American occupational authority (under General Douglas MacArthur) felt that the return of baseball would be a morale boost to the Japanese population. Professional baseball returned for the 1946 season. The uniforms of all the teams included English on them and have ever since. Whether this was something the Occupation forces insisted on or a decision made by the teams themselves is not clear to me.
As Mr. Westbay points out in his response to the original question, Japan is the only one of the three professional leagues in the Far East to use English exclusively on their uniforms.
In Korea, many of the teams have the team name in English on the uniform - but not all of them, at least in 1999:
|99 Teleca #118|
|2000 Teleca #154|
|2000 Teleca #97|
|2008(?) CPBL #119|
|1998 T-Point #120|
|1996 MVP #186|
Interestingly, however, both the Taiwanese and Korean teams for the World Baseball Classic (as well as the mainland Chinese team) have used English lettering for their uniforms, including the names:
|2006 Upper Deck WBC All World Team #AWT-7|
|2006 Flair WBC Moments #CM-4|
|2006 Flair WBC Moments #CM-30|