I'd been on anti-depressants for a bit over a year before starting T, after a major bout of depression in which I became seriously suicidal, and I'm still on them. Just (finally!) owning my gender ID made a huge difference to my mood, and seems to have made me generally a nicer person, but since starting (not quite 3 months ago), I've noticed that I'm a lot more energetic and alert, and I need a bit less sleep.
My clinic, a very good one, does a lot of psychological testing as well as therapy before starting anyone on T, and they're not that worried about mental illness as such, as long as it's being adequately treated. The only thing they really worry about is any tendency to overt aggression. Plain old anger, not so much -- they seem to grasp that some of us have had a good bit to be angry about. I go to a transmasculine group there, and one of the guys has borderline personality disorder, and one of them is bipolar -- non-issues in terms of their starting T.