From my standpoint, the biggest risk is a certain neighbour country with historical appetite for "preserving it's territorial interests".
1) US citizens are taxed on global income forever. (Perhaps it's hard for the IRS to enforce for people whose assets are in countries that don't share information with the US, but it's not impossible.)
2) UK tier-1 investor visa holders must be outside the UK for fewer than 180 days per year, which means they must be in the UK for 185 or 186 days per year, which means they would be subject to tax on their global income during that period. (But of course once they become British Citizens, they could ensure they stay in the UK for fewer than 183 days per year, meaning they'd only pay tax on their local income (which could be zero)).
Latvia, for instance, is much cheaper and gets you into Schengen with an investor visa followed up by investor citizenship.