I am being threatened by unsavory sorts. I am now living in fear for my life.
I am now living in fear of my life.
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The prescriptivist's answer is:
Fear of xxx is the emotion you feel when you worry that xxx might cause harm.
Arachnophobia is uncontrollable fear of spiders.
Fear for xxx is the emotion you feel when you worry that xxx might be harmed.
I fear for my family's safety.
It is a common error to say in fear of when you mean in fear for.
The descriptivist's answer:
Lots of people say in fear of my life, so it's probably okay.
Both prepositions have always been used - "of" is about three times more common overall, but "for" has been steadily gaining ground for the past century...
I see no evidence of a US/UK difference, nor do I believe there's any conceivable different nuance of meaning. Personally I'm quite used to both forms so neither seems particularly unusual.
But if I stop and think about it, for is more consistent with related usages - "I fear for my sanity" (not "I fear of my sanity"). I'm afraid not of my life or sanity, but of the prospect of losing them.
"Of" works better if the noun is something I really am afraid of, such as my fear of. flying.
Given that support from related usages, I'd expect for to be the dominant form in a few decades.