What were their lives like during this time? Were they still treated well by their neighbors or did they face prejudices similar to what people from neighboring Asian countries faced? Did any of them side politically with Japan and turn against their home countries after Japan joined the Axis powers? Was there any state sponsored actions taken against these non-Japanese citizens?
Are you familiar with Tokyo Rose?
It's a generic name applied to a number of female broadcasters of propaganda during the war. The most famous individual identified as such was Iva Toguri D'Aquino, a resident of Los Angeles and a Japanese-American. She was born to a couple of Japanese immigrants. In June 1941 a family member still residing in Japan got seriously ill and no one in that country in the family was able to care for them. So Iva went to Japan, in her haste she lost her passport. In August she applied to the US Consulate to get a new one so she could return, but it wasn't forwarded to the State Department until January 1942.
She was pressured to renounce her citizenship by the Japanese government, and when she failed to do so he was classified "enemy alien", a designation that disqualified her from receiving a ration card and thereby prevented her from buying anything rationed for the war effort. She eventually found work at Radio Tokyo as a typist and translator. During this period she was able to support herself and even visit prisoners of war, during those visits she was able to even smuggle in a little food.
In 1943 the Japanese government began to use POWs to broadcast propaganda. The man tapped to produce was Australian Army Major Charles Cousens who had pre-war broadcast experience and had been captured at the fall of Singapore. Since Iva had smuggled food to him while he was in the camps, he requested her specifically. After she went on air in November 1943, she and Cousens tried to make a farce of the broadcasts as their handlers didn't have strong grasp of double entendres and nuance.
After the war she was arrested and convicted of one count of treason. She served six years of a ten year sentence before being pardoned in 1977 several decades after she had been released from prison. It was determined about that time that much of the case against her was based on mistaken identity and perjury.
Most of the people in Japan for Business or Missionary work were subject to a variety of arrest from House Arrest to internment in a camp and deported in the first few months of the war. There was a civilian exchange run during this period that saw most repatriated. After things turned and it became too dangerous to ship civilians there weren't many Americans or Brits left in Japan.
Of those that were left generally fell into two groups, very dedicated Missionaries who were able to give the first government sweeps the slip and naturalized citizens. Of the former group, they were classed enemy aliens and denied ration books, but they came out alright as only those who had strong communities to fall back on were able to evade the authorities. Of naturalized citizens there are a couple of high profile examples. Rita Tsurutake, nee Jessie Roberta Cowan, a Scottish woman who married Masataka Taketsuru, founder of Nikka Whisky was the subject of a TV drama that ran in 2014 and Gwen Harold, who married Japanese diplomat Hidenari Terasaki, her wartime experience in Japan has been dramatized several times, including the English language book Bridge to the Sun and an 1961 MGM movie of the same time.
As other Axis nations were defeated their citizens in Japan were placed under house arrest or internment for the duration of the war. This process included Germans.
Those who had it worst were Japanese-Americans who refused to renounce their American citizenship. They were often subject to internment, were denied ration books, and a whole battery of daily harassments.
JG Ballard wrote of his experiences of being interned in his autobiography Miracles of Life. He was only 14 when the war and his family's internment ended but his experiences are related with great clarity and ring appallingly true. Those experiences were also used as the basis for his novel Empire of the Sun. Both are excellent if somewhat traumatising reads.
Edit: It appears a South Bank Show Uk arts programme on Ballard is available on YouTube and elsewhere. He doesn't spend a lot of time talking about that period of his life but does touch on it. It's an interesting bio piece of an important author in it's own right.