are we seriously operating under the assumption that the United States government would execute him?

Legally, UK and EU courts must consider this, because sending someone to a country where they will be executed for their crimes is a breach of human rights.

By the strict reading of the law, he could be extradited for life in prison. If he was being extradited to be sentenced to death, that would be a no go.

The US are skirting and pushing the bounds of UK law here. Unfortunately, they will likely get away with it, because the English are pussies.