The Courts after properly considering all the evidence AND fully considering the evidence regarding the risk to the individual.
To leave it to The Home Office/Justice Department would risk politicians using it for their own ends. For example, the cronyism between The US and the current government would mean that any extradition request from The US would be granted as a matter of course instead of the evidence being properly 'tested' in Court before extradition was granted/refused.
A major issue here, however, is political asylum. Even if the Courts fined sufficient evidence regarding the charges, they decline to send someone back to their place of origin to face trial if they are likely to suffer torture/death in their own country. Fair enough you can argue BUT what if the offence is one which carries the death penalty in that country??
So, basically, Mr X is wanted by his own country on, say, terrorism charges which, if found guilty, carries the death penalty in that country. The Courts here decide there is proper evidence to try him over there but they WON'T return him as he could face torture or death.
If he was found guilty then he would face the death penalty anyway. Seems a bit of a moot point to then refuse extradition as his life may be at risk.....
IMHO anyway...