Don't think that it is the same surgeon as most organs are dashed usually by motorcycle courier or ambulance to where the recipient isRegardless of weather you are registered or not the final decision is made by the person's family. We donated my mum's organs which I think is a great gift for someone to do. I am not registered but my family know if something happens to me and anything can be passed on to help others then they can take what is needed. It's a conversation most people should have with their partners and family.
Legally if your registered and the family refuse they could go to court but it's not going to happen. They have a small window to remove organs and get them to the person that needs them. It's normally the same surgeon who removes and fits the organ if I remember correctly