FlashFyre is the only person who can really answer your questions, but he recently left on a trip to India.
When you choose one of the "Big Six", then that will be the nationality of your HQ. Using build points during the campaign to change your HQ to another nationality might have an effect, but I'm not sure about that.
Japan and the USSR don't have allies. Indeed, the USMC might also be considered to not have allies either. I'm not sure what FlashFyre intended when he mentions the Big Six in the first phrase, and then allies in the second phrase. My guess is that he really only intends this campaign format to be played with one of the Big Four (or Big Three) that do have allies.
Next question then becomes, which allies go with which Big Four (or Big Three) nation? Again, I think only FlashFyre knows for sure what he intended. Obviously, even if you go with "historical" allies for the Big Four (or Big Three), you still can get some pretty weird situations. For instance, you choose the US Army in December 1941. Historically, the US Army has as "allies" (I'm defining "allies" as those nations that the Big Nation supplied and/or equipped): Philippines, China, Free French (I'm sure of these, there may be others). Let's say you pick China. Now, the campaign allows the US Army to fight in Europe and Pacific Theaters. So, you could end up fighting with a Chinese core in North Africa in 1942/43 and then Europe in 1944/45. Heh. My personal belief is that FlashFyre didn't have time before he left on his trip to fully explore all the implications and ramifications of his idea here. We'll have to wait for FlashFyre to return before we get a full and comprehensive answer here. For now, your guess is as good as anybody else's.
Also, I personally believe that FlashFyre intended that you only have ONE "minor" ally in your core (I can't imagine Hungarian and Romanian units in the same core, for instance).
Here are some combinations that I think can work okay.
1. Choose Germany and start the campaign in July 1941, and then pick ONE of Italy or Romania or Hungary or Czechoslavakia or Finland core. Fight your campaign on the Eastern Front. There are problems here, but not horrible.
2. Choose US Army and start in October 1942, and then pick Free French core. Fight in Europe Theater. This should work well.
3. Choose Britain and start in May 1940, and then pick Canadian core. Fight in Europe Theater. This should work well.
4. Choose Britain and start in May 1941, and then pick EITHER India or ANZAC core. Fight in any Theater you wish. This should work pretty well.
5. There might be other combinations that work okay, but I'll leave that exercise for you.