In this question: What happens when Wolverine's head is removed? the following panel appears:
Does it imply that that Wolverine can speak without lungs. Is this a super power or some other method of communication?
The panel in your question is from Ultimate Wolverine vs Hulk #5. There are a couple of facts to be kept in mind before drawing conclusions:
There is some machinery attached to his head:
This machinery may or may not be providing him with air to speak.
His lungs are away. How does he breathe? With his skin.
I think it's most likely that the machinery (all the tubes) is feeding him air that allows him to speak, and maybe keeps his blood circulating for him.
One other possibility is that those headphone-like things are reading his mind and feeding it to speakers, but that wouldn't explain why Nick Fury had to interrogate him to learn something, instead of just extracting what he needed.
Yet another possibility is that somehow, Logan was able to adapt and speak without lungs, but that wouldn't explain the need for all the machinery.
Probably by the same mechanisms used to keep him alive in the movies. In the movies, Wolverine lives for centuries because his body is able to regenerate itself after every possible trauma. We get to watch the cellular regeneration in the movies as bullets are removed and tissues are created to replace what was destroyed. He also has super human strength. So it is reasonable to assume that the same mechanism works in the comics. So the likely answer is that he grew new vocal cords along with the strength to use air that his severed trachea is able to pull from below outside his body.