"Tunneling" refers to the process by which a particle may pass through a classically-forbidden potential barrier. It is like bowling, and finding your ball in the next lane over. You didn't throw it hard enough to hop the lane barriers, but there it is.
> particles can exceed c0 (the speed of light in a vacuum) during tunnelling
No particle's speed is measured here, so it cannot exceed c.
What does "during tunneling" mean? Tunneling means measuring its position here, and then over there. You can't do that simultaneously with measuring its speed in between, because of the uncertainty principle.
The wavefunction can be thought of as a probability distribution. That can be updated instantly, even if it becomes very bimodal, because it's just a change in knowledge.