Algorithm to calculate $Wh(G)$ for finitely presented group $G$? Let $G$ be a finitely presented group.
Are there any algorithm to calculate whitehead group $G$, $Wh(G)$ in terms of presentation of $G$?
 A: There is no algorithm. Indeed, the property that $Wh(G)$ is trivial is Markov that is there exists a finitely presented group (${\mathbb Z}$) with trivial $Wh$ and there exists a finitely presented group not embeddable into any group with trivial Wh (any finite non-trivial group).  
 Update Recall that Markov proved undecidability of every Markov property for semigroups (see http://iopscience.iop.org/0036-0279/19/3/M05/pdf/0036-0279_19_3_M05.pdf), and later Adyan and Rabin proved it for groups (see Lyndon and Schupp, Combinatorial group theory, Theorem 4.1, Chapter 4).  
A: 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.
