Btw, replying OP’s hypothetical scenario, Argentina is a very attractive destination. You can get an Argentinean citizenship after living legally 2 years only (and the term “legally” is quite generous here, permanent residence visa is not a prerequisite). Tax wise not very attractive, but if you need a new citizenship and a new passport it is one of the strongest ones you can get it express without any investment requirement.
(But once you have the Argentinean citizenship you can get an Uruguayan permanent residence certificate in a month, which it does grant a 100% tax exception for foreign passive capital income for 5 years)
Public healthcare in Argentina is free and excellent (although its bureaucracy is hellish) and private healthcare is world-class and very cheap.
And once you are an Argentinean citizen, you can apply for a Spanish citizenship by living two years in Spain (taking advantage of a decree that allows citizens of ex-spanish colonies to get a faster naturalization)
PS: btw, extradition-wise, all of Latin America has extradition treaties with both the UK and US.
If you are looking for countries that don’t have extradition treaties with either country and also the best living standards: Kuwait, Qatar, China. The bad news, the first two countries you must live more than 20 years to even consider getting a citizenship, and the latter it is practically impossible to get a citizenship.
Cape Verde in Africa is the next in line that has a reasonable standard of living (it is the most developed country in the African continent), citizenship is achievable in 5 years of residency (quite a standard requirement), citizenship is granted by marriage without residency requirement, and dual citizenship is allowed.