Syria was less of a threat than it had seemed. Apparently, when the Germans saw Rashid Ali's coming collapse in Iraq, they pulled out of Syria, hoping to remove any pretext for a British attack. Syria could wait until they had achieved victory in Russia. They had expended their airborne forces in Crete, and could not use them in Syria.
The British occupation of Syria took five weeks. The French in Syria hated the Free French, and shot at them, even when there was supposed truce, and abused prisoners. The British forces opposing the Vichy troops were an odd-assortment of units, with no cohesion. The air support at the start was meager:
No.11 Squadron (Blenheim IVs), short on aircraft and crews
No.80 Squadron (Hurricanes)
No.3 Squadron RAAF (requipping with Tomahawks)
No.208 Army Cooperation Squadron (one flight of Hurricanes)
X Flight (Gladiators)
The Vichy airforce was much larger. They had 30 bombers and 60 fighters, and these were quickly doubled with arrivals from metropolitan France.
General Wilson commanded the operation. The French were led by General Dentz, the governor. He was located in Beirut, so General Wilson decided to mount an attack there. The 7th Australian Division attacked on the coastal road and on the central road. C Battalion of the Special Service Brigade was attached to the 7th Australian Division. There were also additional units from the 1st Cavalry Division and 6th Australian Division. The right advance was spear-headed by the 5th Indian Brigade, battle-tested veterans. The Free French would follow them. The naval forces included "the 15th Cruiser Squadron (Phoebe, Ajax, with the Coventry, the landing ship Glengyle and eight destroyers). The Vichy French had the destroyers Guipard and Valmy at Beirut.
This is based on the account in Vol.II of the Official History.