I can't resist referring to the paper 

R. Brown and P.J. Higgins, ``On the connection between the second relative homotopy groups of some related spaces'', {\em Proc. London Math.  Soc.} (3) 36 (1978) 193-212, 

which gives some explicit calculations with pushouts of crossed modules, but which is little referred to, and further examples are in 

(with C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72. 

The key to the proof of the theorem is the notion of homotopy double groupoid of a based pair. 

The paper arXiv:0909.3387v2 gives applications of higher van Kampen theorems to homotopy groups of spheres.