There is no algorithm to decide from a finite presentation of a group $G$ whether $\mathrm{Wh}(G)$ is zero. See Corollary 5.7 of this paper which studies various computability issues in higher-dimensional topology.
I think the original question that asks whether one can compute $\mathrm{Wh}(G)$ from a presentation of $G$ does not quite make sense because $\mathrm{Wh}(G)$ need not be finitely generated, e.g. Example on page 2 of this paper gives a virtually $\mathbb Z^3$-group whose Whitehead group is infinitely generated.
For finite $G$, the original question does make sense because the group $\mathrm{Wh}(G)$ is finitely generated (thanks to a theorem of Bass; see again Oliver's book mentioned in comments), and abelian (Whitehead groups are abelian by definition), and isomorphism problem is solvable for finitely generated ableian groups. However, the paper mentioned in the first paragraph above shows that even for finite groups there is no algorithm. On the other hand, a lot is known about Whitehead groups of finite groups, see again Oliver's book.