Can an underactive thyroid and an ovarian cyst (10cm)have anything to do with each other, also period 1 week late, could either of these be causing this?

Yes. Underactive and untreated hypothyroidism can cause late or delayed menstrual cycles. The thyroid has no relationship, however, to the formation of an ovarian cyst. Your cyst, at 10cm, is at risk of rupturing or turning on itself, called torsion. Have you spoken to a doctor about having the cyst drained or removed? No roller coasters, i've seen patients with big cysts get torsion from a coaster.