I'm not convinced the Iraqi revolt will actually happen TTL. The conditions that led to it are very different. OTL Greece had fallen, Crete had Fallen, Rommel had run roughshod over the British in North Africa and Bismarck had shut down the Atlantic convoys taking most of the RN to deal with. Britain looked like it was a spent force so the rebels had a chance of success with Axis support. TTL only Greece falling and the Bismarck's death ride are likely so the rebels may well take a wait and see attitude until after the expected offensives in the Western Desert play out.
OTL the "Golden Square" acted internally to Iraq on 1 April 1941... which is before many of the events you mention
e.g. German attack on Greece not till 6 April
In fact, the first British reinforcements from India had landed at Basra by 18 April before any actual movements vs Britain in Iraq
and these first military movements were rather tentative ... threatening rather than forceful attacks.
IIRC the Iraqis still began their
blockade siege of Habbaniya before the end of April and the fighting there was over by early May
though the British columns from across the desert and up the Euphrates took a few more weeks to arrive and totally crush the Revolt
Bismarck did not sail till 23 May, though that
is a month later than planned,
so unless there was a greater degree of coordination with the Fascists than I believe,
his actions and fate were not a factor either way for the Iraqis who were fully committed before he left.
IMHO the OTL chronology suggests the Iraqis acted purely based on their own ambition, at most encouraged by Rommel's successes at El Algeilah
and therefore will probably still begin acting on a similar schedule ... and still be defeated iTTL in a similar fashion
(since Axis aid iTTL will be even sparser than the negligible help possible iOTL)
Caveat: If Rommel is initially repulsed during March/early April iTTL
I
can see the active fighting at Habbaniya being delayed until the issue in Greece is clearly resolved in the favour of the Axis
which may mean there are less active operations in Iraq if the Brtish can mobilise reinforcements as they did OTL
and thereby force the Square to flee without fighting
And given the active cooperation of the Vichy in Syria and Lebanon by providing bases for the Axis bombers sent to support the Siege
I cannot see the British permitting them to remain in power iTTL any more than they did iOTL
If so ... and if Crete holds and if Rommel is stopped further west ...then the Middle East Theater is different by June 1941
which may set butterflies flying further east