The Iraqi Section of the Arab Socialist Ba'ath Party was established in 1951.
Its an interesting idea - I've no idea on Communist strength in Iraq at the time but I think given the make-up of Qasim's regime (minorities, women, non Nasserite left-wing nationalism) it might be good to have Saddam's bungled assassination attempt work. It will discredit the Nasserites, who lack cohesion. Even by 1963 when Qasim was incredibly isolated, a large minority of the Army fought against the Ba'athist coup.
Qasim is shot (bonus if Saddam doesn't get away), confusion - a Nasserite cadre in the Army attempts a coup, loyalists respond and crush it with the Communist organises strikes and sending their members into the street. A colonel sympathetic to the ICP takes control, the *Popular Front of Qasim's regime remains but with the ICP at the centre, due to its numbers (30,000 odd) and organisation. Cue Soviet aide, clamp down on rivals to avoid CIA backed coup, Kurdish autonomy, promises of secularism, etc.
Cuba style the Front slowly collapses in on itself, leaving only the Iraqi Communist Party as sole instrument of society and governance. God knows what happens next - 'Militarist-Leninist' regimes in Asia and Africa don't have a great track record, though hopefully a half-sincere equality policy means a lot of the ethnic horror of Saddam's days is avoided.
Iran gets even more US support, while Arab Nationalism is now stuck in the middle - they will loathe the new Baghdad regime but by now are firmly Soviet customers. Can see them getting less bang for their buck, if they get any at all. Can imagine a limited Iraqi-Syria War, and for Iraqi oil the Soviets would happily drop contracts with the Nationalists. Sino-Soviet split the Chinese might prove happy backers, though ultimately this boils down to less Soviet hardware and/or some imitation Soviet hardware from Beijing.