You have all of these shipping modules:
flatground
flatair

flatgrounduk
flatairuk

flatgroundrow
flatairrow
flatairrow6

Are the first 2 for the US and the next 2 for the UK and the last 3 for the Rest of the World?

If so, then make a Zone Definition for the US and a Zone Definition for the UK and a Zone Definition for the Rest of the World ... then add the Zone Definitions to the shipping modules ...

Now the US can only see the first 2 and the UK can only see the next 2 and the Rest of the World sees the last 3 ...