From what I've heard and read, T makes us just as much at risk as cis-guys, which actually does mean that we are more at risk of health issues than if we don't transition. To an extent your mother is actually right. Men do have a shorter life expectancy than women, and they are at more risk for things, so chances are that T affects us the same way.
My mother was very interested in knowing all of the risks of hormones when I told her I wanted to transition. She was for a long time worried about it, but what really convinced her that if I need to transition, I should, is statistics and articles about how improved transgender people's lives are when they transition and things like that. When my mother finally accepted that I am a happier person and am much less likely to commit suicide if I transition, she was much more open and willing to help.