Could anyone from a country colonized by Britain join the british military back in WW1 and WW2 ?

I well recall my father who was in WW1 mentioning the Seiks and Hindus from India as being involved. With regard to WW2 I question your inclusion of Ireland. Southern Ireland maintained neutrality and even internment camps for both allied and axis service men. Northern Ireland however took part with Belfast being a principal Atlantic port for Allied shipping.

Mesopotamia as a late-comer that joined the British Empire after 1920 unlike the others you mention.

Its status was that of a 'mandated territory' that doesn't have the full impact of full British Imperial rule unlike the dominions Canada & Australia, or the tight Imperialism in India nor the love-hate relationship with a fellow European nation like Ireland with a veneer of religious difference.

Mandated territory was a piece from the 'Spoils' of the war that nthe victor country appropriates. As such it has less responsibility, but with a putative responsibility to groom the territory towards its independence- which task was fulfilled with reasonable indifference. With it being the spoils from the patently Islamic (Caliphate) Othman Empire, Mesopotamia (even this word is "Greek") had all ingredients of Islamic otherness as viewed from Britain (Australians & Canadians are the British in every aspect except their place of residence, while the Irish are very much similar & akin - the difference isn't like between Eskimoes & the Black people, as is made out to be).

Questions like this, with no breadth of imagination are irritating.

I think you may be confused.

Those folks you listed did not join the British armed forces, but rather, fought in their own units which were part of the "Commonwealth" forces. There were some forces from the middle east, but they were mostly tied down simply policing their home areas.

Mesopotamia was never a British colony.

Look up the Arab Legion and Iraqforce.

There were West Indians, New Zealanders, South Africans, Rhodesians, Kenyans, Chinese and so on and so on.

Mesopotamia was never colonised by the British. After WWI and the collapse of the Ottoman Empire, parts of the Middle east became British 'protectorates', but were never colonies.

In times of war anyone from any country can join and fight for which ever side their personal views dictate they should join. You have to remember the two world wars happened a long time ago when the mobilisation of large numbers of troops was not easy.

As far as I am aware Mesopotamia is a name used to describe an ancient civilisation lying somewhere in the middle east and included parts of Iraq, Iran, Syria and southern Turkey. None of which were ever colonized by Britain nor or they part of todays commonwealth.

Outside of personal choice why would people from that region want to fight on the side of Britain like those from other countries you mention who were part of the British empire?