No, his crime , according to british law, and id bet US law as well, occurred in Britain.
In the hitman example, yes, he would face trial in Britain. That is where the crime of conspiracy to commit murder occurred.
And the crime is not the same in the US and Britain. In one place the criminal penalty is a few months to a few years, in the other it could include the death penalty.
The Extradition is an unjust and I would bet, illegal extradition.
He commited a crime in the UK against the USA. Its no different than throwing a rock at the US embassy. The USA is the defendant and as a UK citizen in the UK in violation of UK law he should be tried in the UK.
If I slap the US ambassador in Canada, I would expect to be charged in Canada under Canadian law. Not extradited to the USA because the victim is a US agent.