If Mosul is added to the new Turkish state as of 1920 (by treaty) or 1921/2 (annexed by Ataturk's army and then recognised by the Powers as they can't stop it, and the UK is glad to be rid of a rebel-prone area) we end up with more Kurds within the borders of Turkey facing marginalisation and being forced to call themselves 'mountain Turks' and learn Turkish by the latter's govts . This could make any 1930s revolt against Ataturk or any 1970s guerilla struggle larger and more dangerous for the Turkish govt, but presumably the urban elites of Mosul and other towns and any upper Tigris valley Arab tribal settlers would ally themselves to the Turkish govt out of fear of a Kurdish state seizing their land and oil. The old urban Ottoman elites in former Ottoman cities in Syria, post-1920 'autonomous' but really run by France, set up local moderate political parties defending the economic and socially conservative status quo in OTL and kept this up after independence in 1943/6 until the more radical military took over in 1949; presumably the conservatives in Mosul would rally to Ataturk or form a separate 'loyal opposition' party in the Turkish Parlt if he permitted this, as less dangerous to their power than struggling for a separate state or joining the poorer, rural, culturally different Kurds. They could not logically link up to Iraqi irridentists within Iraq and try to break away to join Iraq; the Turkish military was much stronger and the UK ran the Iraqi military and supervised the Iraqi political establishment in the 1920s-30s.
Presumably the UK would cut a deal with Turkey to take over the oil wells and run them, similar to that with Iraq in OTl, but would not get such good terms as Ataturk had a large army and stronger will to defy them and could walk out of talks and ally to the USSR. The UK would still need to supervise Iraq, as a stopping-point on the air route to India and to keep an eye on Iran , the USSR control in Baku, and the Arabs in the peninsula; but it would face a weaker Iraqi state if the latter had no or little oil and it might seem less of a high-handed imperial presence 'looting' the country's assets. Would this make anti-UK feeling in the 1950s and the junior military resentment, leading to the 1958 coup, a slower process? The Sunni sector of the Iraqi state would be smaller and poorer if it did not have Iraq's N corner plus the city of Mosul, so its dominance of the Shi'a peasantry of the lower part of 'Mesopotamia' E of Baghdad would be more precarious - but the Baghdad elite and the (Arabian peninsula, from Mecca) Hashemite monarchs would still back a total Sunni control of the state. Presumably the pastoral and 'small town' Arab part of the Sunni coalition would be stronger than in OTL if there was no urban Mosul elite participation in the army and govt; would this make the state weaker both pre and post 1958, and would it affect the make-up of the squabbling 1958-68 military-led oligarchy? Plausibly the weakness of the Iraqi state would make its elite more vulnerable to influence from Syria in the years of Ba'athist control after 1968, or the Ba'ath party would not lose out to the military in 1963-4; and a weaker but still Sunni-led state under Saddam might be even keener to get extra oil wealth by grabbing Kuwait, earlier. The Iraqis might also lose easier to Iran in the 1980-8 war as they had a weaker army, and the Iranians swarm over lower Iraq with rebel Shi'a help in the mid-1980s and force the Iraqis to appeal to the US and the Saudis for help. An earlier 'Iranian revolutionaries vs US' confrontation over who runs Iraq?