We don't have a register, but I've had a donation card since my mid-teens (so early that my Mum had to sign it as my next of kin, I wasn't "allowed" decide it myself yet then). I don't think my parents actually have cards, but there was no issue with signing mine. (I have ticked for everything except my lungs - asthmatic - and eyes - I need glasses, but I get away without wearing those so if they actually wanted my eyes, they could take them).
DH has also had a card for donkey's years, also since before he met me. He's ticked everything (he needs to wear his glasses but otherwise good shape).
DH has his silver pin for blood donation but can't at the moment due to travel to the African continent.
I have tried 4 times and donated twice, but had to go on steroids for asthma in my early twenties so cannot give blood anymore because of that. (If I come off the inhalers, I can, but I DO accept that I need them).
We have talked about it in relation to DD in the past, and if anything ever happened, we will be donating her organs too.
I am currently looking into the option of donating my body to medical science (which would probably rule out donation of organs) - but I am only in the early stages of that and I am hopeful that I'd have YEARS to organise myself....