Tyr had survived because he too, caught on to Noa's suspicion, he caught Noa's eyes following him on multiple occasions ever since the death of Gissur.
Noa was not as subtle as he thought and Tyr confirmed his suspicion when the assassination attempt failed but a few of the prisoners were let go of that night alongside Noa's disappearance.
Tyr knew only one was responsible and if they had multiple targets, then he would understand why but Noa was nowhere to be seen.
Tyr knew how he got Noa and was prepared to lose him that same way but Tyr also saw an opportunity in this betrayal.
He kept it all to himself and allowed everyone to carry out their plan as intended to give whatever information Noa was about to give Sighvatur credibility but to also confirm if he indeed did betray them.