I don't see how citizenship is a major stumbling block; any ex-president eying the premiership could simply move to the UK, become resident, and apply for citizenship. The only legal requirement to be PM is that one be a British (or Irish, or Commonwealth) citizen, there's nothing about how or when the citizenship was obtained.
No, the real problem here is optics and politics. Any former president is going to be seen as a symbol of America and American power, which are not appealing qualities when trying to lead a different country. Their campaign would be blasted as the zenith of "Americanisation" of British politics at best, and as a foreign agent who wouldn't stand up for British interests and values at worst. I can even imagine it being portrayed as a "reverse colonization". And that's not even getting into how their record could be used against them— if the president had ever sided against the UK in some international issue, that's not going to play well (to say the least).
For this to happen, you'd need a very specific set of circumstances. First, there's going to have to be a long period of time between their presidency and premiership, for the politician to effectively reinvent and recast themselves as truly British. Second, their presidency must be inconsequential enough that it won't weigh on them so much.
So here's a possible scenario. Bill Clinton is never accepted into Yale (or just chooses not to accept the offer) and instead stays at Oxford, where he additionally meets a woman whom he marries. After he completes his degree, Clinton moves back to the US with his wife/girlfriend, where he gets into politics and his career goes more or less the same (Governor of Arkansas, 79-81, 81-92). Come the 1992 presidential primaries, he mulls it over but ultimately sits it out, but is later selected as victor Tom Harkin's running mate in the name of balance. Harkin/Clinton win election, but Harkin dies sometime in after the 1994 Republican wave, and Clinton ascends to the presidency. Clinton, who is more to the right than Harkin, is more willing to work with Newt Gingrich, which incenses his own party; Clinton is ultimately denied renomination in 1996. His political career over, Clinton and his wife move back to the UK. There, Clinton finds his way back into politics via a series of appointments from Tony Blair (ambassador? I don't know), eventually joining the Labour Party and standing for election as an MP in 2005. Then you just need to have Clinton win the leadership and an election, or for him to serve as PM on an interim basis following a death…