Individual nanites don't hold that kind of information. It's more of a combination of The Network and Matrix-aligned "Cloud computing" ... each nanite has a teensy nugget, which is similar to a single brain cell. When a critical mass is reached, they can communicate with one another and start having a semblance of a mind. This give sthem enough computing power to move up to the brain and start a re-write, downloading a teensy part of the personality as they "re-code" the brain. The more nanites present, the faster this rewrite can go. In essence, you're moving water from a lake to a bucket one eyedropper at a time. Once the full rewrite's done, the entity's no longer in the Matrix at all.
(Or, rather, that particular copy isn't. Each individual Monad has been copied, or partially copied, several dozen times now.)
Now, the BIG problem is when naites from two different Monad clusters are both in a body. If the "Steve" Monad and the "Kevin" Monad are both inside the body, they're each trying to code the brain with their sourcecode, overlapping one another's work. You can see what happens in this case in Boston.