I would like to say that since one of my colleagues left, nobody cares about our project anymore. Or nobody has cared? EDIT: I want to say that nobody cares now and it started when X left.
I'll disagree with some of the other people and say:
He was the last person to care about the project, and since he left there has been nobody who cares.
Saying nobody cares is a statement about the present state of affairs: nobody cares now.
If you want to suggest the point in time when it was first true that nobody cares, that was in the past: Nobody has cared about the project since he left. You could even put the two together for emphasis:
English is subtle and malleable, though. If you say since he left, nobody cares about the project I would take that to mean because he left, nobody cares about the project.
Both are possible, depending upon what you want to say. Are you talking about what happened since he left, or what is happening?
Edit: per OP's edit
"Nobody cares" is correct if nobody cares now.
"Nobody cared" would be correct if the lack of caring was in the past and either the incident is largely forgotten or people care now, despite not having cared in the past.
"Nobody has cared" would most commonly be used if no one cared in the past and the state of caring is potentially changing: "Since Jim left, nobody has cared about the project, but we're hoping to revive interest."