I don't know how they would deal with someone if their family objected and they were legally domiciled elsewhere. The card you carry though, has a statement saying that your wishes are to be a donor, and then there is a place to sign. You also list a contact number for someone to be notified in case of emergency. (I have my sister in PA listed.)
According to the Donor website:
What will happen if my relatives object?
We know that in most cases families will agree to donation if they knew that was their loved one’s wish. If the family, or those closest to the person who has died, object to the donation when the person who has died has given their explicit permission, either by telling relatives, close friends or clinical staff, or by carrying a donor card or registering their wishes on the NHS Organ Donor Register, healthcare professionals will discuss the matter sensitively with them. They will be encouraged to accept the dead person’s wishes and it will be made clear that they do not have the legal right to veto or overrule those wishes. There may, nevertheless, be cases where it would be inappropriate for donation to go ahead.
My non-professional interpretation of this is that in the UK, it's the wishes of the donor that override that of the family.
Perhaps someone with some experience in this may chime in?