Ramadi was occupied by a "considerable force of Iraqis", and there was another one south of Fallujah. Since General Clark preferred not to have these in his rear, he hoped they would surrender, if they were bombed. However, that was not sufficient.
On 28 May 1941, Brigadier Kingstone's force had advanced quickly, as they were on a hard, "flinty" desert plain. They had to fight an action at Khan Noqta, and then "a few miles further on". They found a working telephone, and an interpreter spread misinformation, hoping to panic the Iraqis. They ended the day 12 miles from Baghdad, but facing a blown bridge and having a canal to cross.
This is based on the account in Vol.II of the Official History.