For most of these, I think they lump the (now freestanding) countries in to their colonial domains: e.g. Singapore owned by UK, Burma being a "Shan State" of the UK, New Guinea owned by Australia, etc.
But, yeah, you're dead on and I always like the more broken-down ones.
"British Raj, British Malaya, the Philippines and Finland are not included." Why not? They were all involved. As were Netherlands and Belgium, why are they missing? And PNG?
Germany
Italy
Japan
Co-signers of the Tripartite Treaty:
Bulgaria
Hungary
Romania
Slovakia
Countries that were annexed by, or at war with, Axis Powers before the World War II:
Austria
Ethiopia
Republic of China
Allied Powers:
Australia
Brazil
Canada
Newfoundland
New Zealand
South Africa
Soviet Union
United Kingdom
United States
Countries that were attacked, occupied, or switched sides during the war (Most countries below had declared their neutrality before being assaulted.)
Algeria
Albania
Belgium
Burma
Czechoslovakia
Denmark
Estonia
Finland
France
Greece
Iceland
India
Iran
Latvia
Lithuania
Luxembourg
Morocco
Netherlands
New Guinea
Norway
Philippines
Poland
Singapore
Syria
Thailand
Tunisia
Yugoslavia
But, yeah, you're dead on and I always like the more broken-down ones.