So the death penalty does not exist in the UK, and correct me if I'm wrong, but I beleive our laws also state that a person may not be extradited for prosecution if the conviction of the crime will result in the death penalty under the foreign country's laws. So considering the above, what would happen to a person who commited a serious crime that would result in death penalty in the foreign country, and is also a serious crime here in the uk (murder, sexual assult), could they be prosecuted and imprisoned in the uk, following uk laws?
Genocide, crimes against humanity and war crimes can be prosecuted in the UK no matter who did it or where. Murder and rape are not included in the Universal Jurisdiction offenses. There are some other crimes that can be extraterritorially prosecuted (see here) covering bribery, money laundering, tax evasion, circumvention of trade restrictions, but this does not include murder or rape.
Therefore if, for example, Jones is accused of murder in Texas, the Texas prosecutor must give assurances that the death penalty will not be sought. If he won't give such assurances, Jones cannot be extradited from the UK, and he also cannot be prosecuted in the UK for that crime.