The captured survivors were spared from execution, assuming Sela is telling the truth
The only person I found that refers to the Federation survivors of the Battle of Narenda III, is Sela. Although the situation between Jean-Luc Picard and Sela was tense, and misinformation might have been useful for Sela, I will assume that her statements regarding the crew of the USS Enterprise C are true.
From Redemption, Part II:
SELA: Yes, she was on that ship twenty four years ago. She was sent there by you from the future. She was among those few who survived. They were all to have been executed after the interrogation, but a Romulan general saw her and became enamoured with her. So a deal was struck. Their lives would be spared if she became his consort. I was born a year later.
PICARD: I want to meet your mother. Can you arrange that?
SELA: One night, when I was four years old, she came to me. She bundled me up and she told me to stay quiet as we left the compound. I realised she was taking me away. She was taking me away from my home, my father, so I cried out. My father offered her life. He gave her a home, gave her a child, and how did she repay him? By betrayal. They executed her. Everything in me that was human died that day with my mother. All that's left is Romulan. Never doubt that.
From this quote we know that:
There were other survivors than Tasha Yar.
They were interrogated by Romulans and they were supposed to be executed after that.
Tasha Yar struck a deal with a Romulan general, so that the lives of other crew members will be spared.
We do not know if the Romulans kept their
part of the bargain, but assuming they did, the crew has not returned
to the Federation, as their fate remained unknown to the Federation.
We do not know if they were sentenced to life inprisonment (in place
of the death penalty) or home arrest, or anything else.
We know that
Tasha tried to escape with Sela, but it failed. Tasha was executed
for that act, so we have a confirmed death of a survivor after 4
years. We do not know anything about the fate of the rest of the
crew, but as far as we know Romulans, even if they had kept their
part of the bargain initially, they might have voided it due to
Tasha's actions. These are only assumptions, of course.
It is mentioned in the Vulcan's Heart non-canon novel that:
The only members of the Enterprise crew to survive and also avoid Romulan capture were Doctor Frances Stewart, two security guards, and a shuttle pilot, all of whom were assigned to accompany the stricken Saavik to Vulcan. They survived because their shuttle left the ship before the Romulan attack took place.
Source: Memory Beta
Therefore, we can add another part of the crew that avoid being captured. As the question refers to the survivors of the USS Enterprise-C (not the survivors of the Battle of Narenda III), I think it should be also mentioned in the answer, although these crewmembers are unaware of the events mentioned above.