Praise be to Allah.
There is nothing wrong with working as a midwife, for free or in return for payment. It is work that benefits many Muslims, so there is the hope of reward for those who have that intention.
There is nothing wrong with studying and working in the hospital for the period that will qualify you to become a midwife. What you mentioned about being allowed to wear hijab and pray is a good thing.
With regard to the uniform, you must leave your house wearing hijab and covering clothing, then you may put on the uniform in the hospital.
With regard to uncovering the forearms or part of them, that is not permissible if any non-mahram man is present, and you must try hard to cover them, even by wearing medical gloves, and putting another pair of gloves under them, and the like, so that none of your forearm will be visible. The fuqaha’ are unanimously agreed that the woman’s entire body is ‘awrah, although they differed concerning the face and hands. But whatever is above the hands must be covered, according to scholarly consensus.
According to the most correct view, the hands must also be covered. See the answer to question no. 45869 .
If there are no non-mahram men present, then there is nothing wrong with uncovering your forearms.
And Allah knows best.