Personally I think it's a great idea.
What needs to be borne in mind is that (a) a person can opt-out by signing up to a register; and (b) the next of kin can block organ donation if they have a problem with it.
So there's no real chance of having your organs harvested should it be a problem with either yourself or your loved-ones. Countries which have adopted this policy have -- overnight -- ended the shortage for most organs and slashed waiting lists.
If there is a problem, it's going to be the financial cost. Transplant surgery and rehab is extremely expensive. If the NHS finds itself performing (let's say) twice as many transplants per annum then it'll have to either get additional funding from the government (unlikely) or else divert funds from elsewhere within the NHS (more likely).