Her page on memory alpha doesn't have much biographical info, but it does say "She ran the hundred meter dash in record time", and this seems to be based on the animated series episode "The Slaver Weapon"--according to the transcript, she said "I'm slowing down. I used to run the hundred in record time." Assuming the original Enterprise didn't have a big running track (it generally had a more cramped, submarine-like feel compared to the Enterprise-D in TNG), this would probably refer to some time before she joined the five-year mission.
You could also say her comment in "The Trouble With Tribbles" here, which aired later in the same season as "The Changeling", suggests the likelihood that her memory extends beyond just the very recent past:
KIRK: I see you didn't waste time taking your shore leave.
UHURA: (coldly) How often do I get shore leave?
That's not a comment she'd be likely to make if she had no real idea of how often she had been getting shore leave prior to "The Changeling", though I supposed she could have checked her personal records.
One last point is that scene where they learn of Uhura's memory loss in "The Changeling" goes like this:
KIRK: Repair that unit.
NOMAD: Not possible.
KIRK: You restored Scott. He had much more extensive damage.
NOMAD: The unit Scott required simple structural repair. The knowledge
banks of this unit have been wiped clean.
SPOCK: Captain, if that is correct, if there has been no brain damage
but only knowledge erased, she could be re-educated.
KIRK: Bones?
MCCOY: Yes. I'll get on it right away. Oh, and in spite of the way you
repaired Scotty, you metal ticking—
SPOCK: Does the creator wish Nomad to wait elsewhere?
And this page notes that the James Blish novelization of the episode modified the scene slightly:
"Can you repair her, Nomad?" Kirk demanded.
"Not possible," said the machine.
"But you were able to restore Scott, who had much more extensive
damage."
"That was simply physiological repair. This one's superficial
knowledge banks have been wiped clean."
"Superficial? Be more specific."
"She still remembers her life experiences, but her memory of how to
express them, either logically or in the illogic called music, or to
act on them, has been purged."
"Captain, if that is correct," Spock said, "if her brain has not been
damaged and the aphasia is that superficial, she could be taught
again."
"Bones?"
"Yes. I'll get on it right away." McCoy swung on Nomad. "And despite
the way you repaired Scotty, you ticking metal--"
"Does the Creator wish Nomad to wait elsewhere?" Spock broke in
quickly.
Even though the filmed version is canon and the novelization is not, perhaps we could imagine that Nomad's explanation in the novelization still held in the filmed version--that when Nomad said "The knowledge banks of this unit have been wiped clean", it only intended "knowledge" to refer to certain skills such as language, not to memories of life experiences. Also, "wiped clean" could just mean a scrambling of the neural connections between her knowledge of language and her ability to translate audio and visual signals into words, or her ability to translate words into the appropriate movements of her vocal cords and hands so that she could speak and write, without actually scrambling the parts of her brain that knew the meaning of the words. This would be analogous to the fact that in real computers "erasing" a bit of data does not necessarily mean erasing the data itself, but only the "pointers" that various programs have to the data's location in the memory banks, as discussed in this article.
