This story is of interest to me, with the guy likely to be extradited to the US on fraud charges.
https://www.bbc.co.uk/news/business-60178089
His defence team have stated ‘He is a British citizen who ran a British company in Britain subject to British laws and rules and that is where the matter should be resolved. This is not the end of the battle - far from it’
I’m assuming that the claimed misrepresentation of his company’s worth was made in the UK and if so, that is where the case should be heard IMHO. Otherwise he could be convicted of a crime, that is not a crime in the UK (unless the US court applies UK laws to the case).
If I murder an American citizen in London, I can see no reason why I should be tried in the USA for such a crime – but the arrangement Tony Blair’s government made with the US, whereby they can request extradition of British citizens and we will roll-over and put them on a plane may override that.