hair loss is a symptom of hypothyroidism, so it could well be unrelated to your diet. see this list of symptoms that i googled for:
Weight gain or increased difficulty losing weight
Coarse, dry hair
Dry, rough pale skin
Cold intolerance (can't tolerate the cold like those around you)
Muscle cramps and frequent muscle aches
Abnormal menstrual cycles
if you have hypothyroidism, i'd look into that as a cause of the hair issues before guessing at anything else being to blame. it could be caused by a poor diet, but if you know you have a medical condition which could very likely be to blame, that'd be where i'd start looking. perhaps your medication/treatment needs reveiwing a little? did you mention your hair loss to the doctor who treats your hypothyroidism?
but by the sounds of it, your diet could do with a bit of a change anyway, so it might not be a bad idea to look into that, just from a general health point of veiw. if it helps your hair, great- if not, the rest of your body'll still thank you for it, right?
in my mind, vitamins are good as a stop-gap/top up, but you can't beat a healthy diet with plenty of fruit and veg, whole grains, beans, nuts, etc. tell your mum that you're really keen on improving your diet and health, and volunteer to help with groceries and cooking- thats often a big incentive for stressed overworked mums!
you can always start small- throwing a few steamed frozen veggies or canned beans into your ready meal... switching to whole grain bread and pasta, etc, and working from there. why not think about getting an easy vegan cookery book (i get mine from from the library) and having fun making new things- maybe start with some simple pasta sauces, salads, soups, etc!