Nobody but the Labour party elected Tony Blair Prime Minister. And if they ever get fed up with him they can kick him out - just like the Tories did with Maggie Thatcher.
If he died tomorrow he would be replaced from within the ranks of the LABOUR party.
So, no, the UK voter does not get to elect the Prime Minister - and he is not Head of State, that is the Queen regardless of her level of power or whether you like it or not.
The Prime Minister is just "Primus Inter Pares" amongst MPs, regardless of how he likes to seem to think of himself.
That's just how our system works.