For the most part mental illness can be directly linked to family history of mental illness. Meaning that if your Mother had diagnoses of Bipolar Disorder you, her daughter/son is more likely to exhibit symptoms of bipolar disorder. But only more likely, other things like environment child is raised in and which genetics child obtains from parent can determine likelihood of symptoms developing in child. Age of onset of symptoms of bipolar disorder is mid teens early 20s as far as i am aware.
Also one does not start with having symptoms of ADHD and then end up with Bipolar Disorder, the two disorder are connected like that. Although with some disorders there is a high likelihood that you will have one and the other. For example there is a high rate of individuals who have symptoms of depression AND symptoms of anxiety. We call that co-morbidity, when two disorders have a high chance of dual diagnoses. In the same respect ADHD does not result in schizophrenia.
As someone who has minimal experience in behavioral health and is NOT a doctor I would recommend taking your doctors advice to heart and not worry about what could or will happen down the road. If your child is exhibiting symptoms of ADHD then worry about THAT now, and maybe do some research on coping mechanisms for children with ADHD, for example maybe your child NEEDS to move in order to learn, or pay attention. Personally I know if I am in a lecture I need something to do with my hands otherwise I can't focus. Maybe talk to his teachers and see if it would be ok (if not too distracting for the other children) if he stands in the back, or has something to fiddle with at his desk. Maybe that will help. But like I said, I am not a doctor, nor am I counselor. Please take your doctors advice.