Not to spoil things but yes Britain and Russia will be joining the space race shortly. Other nations, more than OTL, will also be developing their own programs.
Good question on what the Fascist like governments would be called ITTL. I haven't been able to think of a good name yet but would love to here suggestions. As of 1961, Japan is the closest to OTL fascism albeit with an antiwestern/colonialism bent. China is really more of a military led government than actual fascism.
Fascism is more than extreme nationalism and totalitarianism: it's a distinctive approach to economics and civic society in general. OTL Fascism (think Mussolini) drew from secular Syndicalism, while utilizing the language of Catholic syndicalism. Fascist/Corporatist economics seeks a middle ground between laissez-faire capitalism and Communism, seeking class collaboration instead of class neglect or conflict. If Japan can seek a policy as such, we could call her Fascist.