The principal commanders involved n the Greek campaign decision-making were the GIGS, General Dill, General Wavell and Admiral Cunningham. They all had doubts about undertaking the campaign but understood that te politicians thought that the right thing to do was to support Greece. The Prime Minister, Mr. Churchill, so dominated his cabinet and military staff that no one was prepared to oppose what Mr. Churchill wanted to do. It was also true that Churchill was more knowledgeable than his peers in any government involved in the war. One thing that created Churchills dominance was his prestige.
The Australian Prime Minister, Mr. Menzies, was surprised at how Churchill had established his "dominance over his cabinet". There was no one in early 1941 who was able to tell Churchill when he was wrong. When Alan Brooke arrived on the scene, he made it his responsibility to attempt to keep Churchill from doing things which would cause trouble.
The "Dominion Governments" might decided to block the Greek campaign, but the way that Churchill operated in early 1941, he withheld information from the Dominion governments that would have made it possible for the governments to be part of the "strategic decision-making process". General Blamey was told that Mr. Menzies had agreed to the Greek campaign while Mr. Menzies was told that General Blamey had agreed. This seems like a general lack of respect for the Dominions.
This is based on the account in "Greece, Crete, and Syria" by Gavin Long.