Yes, though in the case of South Africa and Australia I believe they were the actual mandatory powers alongside the UK itself. This would be fundamentally different than with a Iraq/India scenario as India itself would be unlikely to be mentioned as a mandatory power in the treaties governing the cession of Mesopotamia and instead the mandatory power would solely be the UK. This would mean that if the administrative union between India and Iraq were maintained through to 1945 then Iraq would become a Trust Territory of the UK by 1947 instead of a Trust Territory of India (as happened with northern New Guinea and Australia and as should have happened with South-West Africa and South Africa except South Africa refused and kept administering SW Africa under the terms of the mandate).
India was at the table at Versailles; the Government of India was essentially at parity of importance with the Dominions, it just so happened that the people of India didn't have much say in its governance. India could be a mandatory power in its own right, and especially so when Arabia traditionally fell under the power of the Government of India- only the High Commission of Egypt started to change that up.