Mainly, my question is in the title, but let me be more precise here.

Let $G$ be a finitely presented group with solvable word problem. If G is not left-orderable, is there an finite-time algorithm to establish this fact?

If the answer to the above question is unknown, is it known in the case $G$ is the fundamental group of a 3-manifold? Or is there a class of groups where it is known?