If a person is charged with a felony by the United States and the United Kingdom, where would he?

It would depend on where the crime was committed. If in the UK, trial would be in the UK and vice versa. If by some wild chance the crime was committed in bothe the UK and the USA (some kind of computer crime for example) then it would depend on, first, the nationality of the person and then a decision on who has juriprudence in the case, that is who has the most to gain (or lose) by taking that person to court, that decision is often/usually political and the province of the relevant embassies.