My gut tells me (but don't have any knowledge to back it up) that your body might have been shocked by the RAI and messed up your period. I always took my BC whether I had started a period or not (but I was taking it to regulate my periods anyways). Is it possible you are pregnant? If it is a possibility I wouldn't start until you had a negative test.
My periods stopped long before I was diagnosed with Graves and Hyperthyroidism and I had RAI 6 months ago.
I went for hormone tests this week as I heard that thyroid disorders can cause fertility to drop and periods to cease or become irregular.
My results showed I am well gone past the menopause!
I am 48 years old and havent had a period since I was 43.I wont get any now either.
I have seen others post that they cannot fall pregnant because of the TSH being out of range (either hypo or hyper).
I would say the best thing to do would be a pregnancy test and if negative, see your Doctor as the RAI can do a lot of things to your body.
Chances are , your periods will resume to normal once your TSH does.
In regards to BC, I would ask your Doctor whether to take or not.