I know Churchill had a lot of influence, but did he have the power to up and fire Brooke over this.
He was able to remove Dill as CIGS, but turning around and replacing Brooke after just a few months may not have been practical. maybe after ten months or a year? There are experts here on British cabinet politics who could clarify this.
I think that at the least it was the Vichy Goverment keeping its options open.
They certainly were. In 1942 the US ambassador to France, Admiral Leahey, found Petains government open to discussions about reentering the war. Darlan was premier at the time & his response to Leaheay is suposed to have been "If you come with three divisions we will fight you, if you come with twenty we will join you." It looks like he was telling Leahey that the Allies needed to invade with maximum force and not some sort of poorly prepared & weak probe. The confusion of the French leaders when Operation Torch was executed shows how divided they were, and how many of them tried to have it both ways. Some like General Barre or Lattre de Tassigny attempted to fight the Germans with their commands in November 1942. Others like Nougues resisted the Allies. To many others like Darlan dithered, hoping somehow nuetrality could preserved even as entire Allied and Axis armies were marching into French territory. The inability of Petain & his subordinate leaders to decisively commit one way or another in 1942 doomed his government to irrelevancy for the remainder of the war.
Yes British troops had fought the Vichy and Churchill disliked their failure to oppose the Germans very much. But if Brooke went around Churchill and convinced the U.S. to support the plan, how would Churchill react? Refuse to allow the operation when American loans float the British economy and American goods float the British war effort? I'm sure if Roosevelt pushed hard enough he would be able to get the invasion, I mean OTL the British postwar felt like the Americans had skinned them with the many concessions forced upon them, so it wouldn't be too hard to see Roosevelt coerce Churchill to accept the invasion. And if Brooke supports it as well Churchill's position to argue is even weaker.
Churchill did prevail in stopping plans for a 1943 invasion. At the January 1943 SYMBOL Confrence @ Casablanca he talked Roosevelt into going against the advice of Marshal and committing to the Mediterranean campaign that Brooke argued for. What would have happened had Roosevelt come down solidly on the side of Marshal I cant say, other than the confrence would have lasted a lot longer ; )
Sorry for not being clear enough in the OP but I intend this to be 1943 invasion, but the decision for the invasion to occur would be in early 1942 so that the U.S. can prepare for the invasion by moving men, planes, and material to Britain.
Marshal thought a decision for a 1943 invasion had been made at the previous joint chiefs confrence (QUADRANT). He had been steering the US mobilization towards a 1943 invasion of NW Europe through all of 1942, despite the emergencies in the Pacific, USSR, and Africa. He was suprised and angered when Brooke came to the SYMBOL Confrence in January 43 in direct opposition, and supported by Churchill. Pogue in his biography of Marshal, Grigg in his analysis of the SYMBOL Confrence, and Atkinson in 'Day of Battle' all examine this pivotal meeting in some detail, including previous US plans.