I know it's a hypothetical question but...
As far as I understand it, there's very few examples where a live donation would be performed outside of immediate family and friends. Sometimes, where a family member is willing to give a transplant but is not a tissue match, it can be possible to do a swap with another donor.
And as for ethnic background, donors from the same ethnic background are generally closer matches, so normally you would donate within your own ethnic group anyway.
After I'm dead they may have what they please for whom they please, and indeed, if you're registered on the register
http://www.uktransplant.org.uk/ukt/how_to_beco me_a_donor/registration/consent.jsp
you must necessarily conceed any rights to dictate who may recieve your organs.