I'm actually writing an essay for my medical law module (LLB Law) on this very topic. Basically, I have to assess whether the current law is satisfactory or whether reform to an opt out system and a market in organs is advisable.
I am all for organ donation -it can save the lives of those who would otherwise die. Once we're dead, we don't have any need for our organs (I believe) -there is even an argument by a guy named Harris that there should be no choice in the matter. After all, he argues, our bodies can be cut open by a coroner if there is any question of fowl play or if we die of an infectious disease/unknown reasons etc. We don't "own" our own bodies in law -because you can't own yourself!
However, the problem many people (viz. academics) have with cadaver organ donation is the definition of death. Your heart can be removed whilst it is still beating. This comes down to the legal definition of death -which is brain stem death. it is thus perfectly possible for a person to be on a ventilator but actually be dead for the purposes of the law. Obviously, organs degrade quickly, so there is every need to act quickly. But things have been tried. For instance, one pilot programme in the 90s saw the NHS undertake "elective ventilation". This is fundamentally where a person who is an organ donor who is dying will be moved from the ward to ITU, put onto a ventilator and allowed to lapse into a coma. Then when brain stem death occurs, their organs would be removed. The problem with this was that people's last moments were being needlessly spent on ITU and not on the ward with their family/invasive treatment etc.
There is also the argument that we should take organs from those in PVS -after all, those who have been in it for more than one year have a very slim-to zero chance of even regaining consciousness....so it's the good old slippery slope argument.
There is also the argument that people should be able to buy and sell organs (the argument goes that the NHS should administer this to avoid the rich queue jumping). Arguments against this include exploitation and risk. But proponents point out that we can do risky jobs (i.e. many people in LDCs work in dangerous jobs. The mortality rate for live kidney donations is less than 1:3,000. Apparently, if one of your kidneys is damaged the other one is also often damaged, so to argue that you might need your other one seems a bit redundant. Also, the argument goes that you can do a "dual good" -i.e. help someone get out of poverty and also save a life/cut waiting lists (given the cost of dialysis on the NHS, the NHS could potentially actually afford to do this)!
So lots to think about!