You don't need nanites; cybernetics exists in falloutverse.
Even if it's cutting edge stuff.
See: old world blues in fallout new vegas.
You could have parts of you, including parts of your brain, replaced piece by piece with cybernetic\synthetic versions.
Eventually you're completely synthetic, including your brain. You get into Ship of Theseus territory, if it's really you now thats everything been replaced, etc.
That can be ignored, or you can have the brain-part replacement optional so people on both sides of that issue can pick
what they're comfortable with.
Maybe someone whose good at brain surgery but can't program synths.
So since they cant make good personalities\smart synths they do conversions instead.
but they can 'copy' pieces of brain and then replace them.
so, copy one quarter of your brain, replace it with synthetic variant.
You're 'tested' and spend a short time as that, then another piece of your brain.
If it's piece by piece then it's still a conversion and a continuation of self, whether true or just experienced.
