I will reword and answer the question as, "How well accepted were the Hashemite Kings in that the British imposed on Transjordan and Iraq after World War I?" Because the answers are rather different.
The current king of "Jordan" is Abdullah II. He is the son of the late (and popular) King Hussein. Hussein was the grandson of Abdullah I, one of the two brothers (along with Ali), enthroned by the British in what was then "Transjordan". The Hashemites have continuously ruled modern Jordan for almost 100 years and four generations, with progressively greater acceptance. Whatever misgivings the TransJordanians might have had about them at the beginning have disappeared. (The British influence that "set up" King Abdullah I has long since been gone, so what has happened since then is quite "natural.")
The "transplant didn't work nearly as well in Iraq. That country experienced several governments between the world wars, including a pro-Nazi government. After World War II, the Hashemites tried to regain power in Iraq with limited success. In 1958, the Iraqi (Hashemite) monarch proposed to Jordan's King Hussein that the two countries merge their monarchies as a counterweight to the Egypt-Syrian United Arab Republic. This was met by a coup led by Qasim, an Iraqi nationalist who overthrew the Iraqi monarch. (He was in turn overthrown by the Baathist party in 1963.)
There was basically no group in Iraq that supported the Hashemites. Certainly not the Shiite Moslems (the Hashemites are Sunnis), nor the Kurds, who are fellow Sunnis, but had their own agenda. Only British support enabled the monarchy to last as long as it did in Iraq.
So I would say that the Hashemite kings were considered much more "foreign" in Iraq than in Transjordan.