Itching nipples, particularly the areola, are a common side-effect of hormone therapy in trans women; and may also occur in trans men. It is usually treatable using a hydrocortisone cream, which is available in most chemists, or even simple moisturiser for milder cases.
Itching and dryness of the skin is a symptom of eczema, which can be caused or exacerbated by hormone treatment. The skin often becomes dry, cracked and itchy and will be made worse by persistent scratching or rubbing; neither of which will remove the itch.
Treatment is commonly in the form of a corticosteroid or hydrocortisone cream, which will heal the damaged area as well as soothe the itching. If over-the-counter medications do not work, consult your GP.