This is a belated reply to this, but I had no time to look at mathoverflow while finishing the book on "Nonabelian algebraic topology:..." .

First it should be emphasised that we get strict fundamental crossed complexes and $\infty$-groupoids (cubical or simplicial $T$-complex) for filtered spaces, not for a "bare" space. Indeed the fact that this strict theory works for filtered spaces, and so generalises the usual fundamental groupoid of a space with a set of base points, is part of the case for filtered spaces as a basic concept in algebraic topology. It will take time for this penny to drop!

The simplicial nerve of a crossed complex gives the usual nerve if the crossed complex is simply a groupoid in dimension 1.

The simplicial nerve of a crossed module is described in dimensions 1,2,3 on p. 323 of that book. See also the last part of section 3 of

`Groupoids and crossed objects in algebraic topology',
*Homology, homotopy and applications*, 1 (1999) 1-78.

for a crossed module $\mu: M \to P$ the nerve $K$ is given by:

$K_0 = {0}, K_1=P$. The 2-simplices of $K$ are quadruples
$k=(m;p,q,r)$ such that $m \in M, p,q,r \in P$ and $\mu m = q p r^{-1} $
with $\partial _0 k=p, \partial _1 k=r, \partial _2 k=q$.
The 3-simplices of $K$ are quadruples $(k_0,k_1,k_2,k_3)$ of 2-simplices
such that if
$k_i=(m_i;p_i,q_i,r_i), i=0,\ldots,3$, then
\begin{equation} m_0^{p_3} m_2 m_1^{-1} m_3^{-1} =1,
\end{equation}and the edges of the
2-simplices $k_i$ match up to form a 3-simplex.
For $n\geqslant 4$, an $n$-simplex of $K$ is an $(n+1)$-tuple of
$(n-1)$-simplices of $K$, whose faces match up appropriately.

Actually in the reduced (one pointed) case this construction of the nerve of a crossed complex (but called a "group system") is essentially in

Blakers, A. Some relations between homology and homotopy groups.
*Ann. of Math*. (2) 49 (1948) 428--461.