Let me write $\mathcal{N}_g$ for the mapping class group of the connect sum of $g$ projective planes. Nathalie Wahl proved that these groups enjoy homological stability, and in

O. Randal-Williams, The homology of the stable non-orientable mapping
class group, Algebraic & Geometric Topology 8 (2008) 1811-1832.

I calculated the stable (co)homology. Using the best currently available general homological stability ranges, one has
$$H_1(\mathcal{N}_g ; \mathbb{Z}) = \mathbb{Z}/2 \quad \text{for} \quad g \geq 7$$
and
$$H_2(\mathcal{N}_g ; \mathbb{Z}) = (\mathbb{Z}/2)^2 \quad \text{for} \quad g \geq 10.$$

In fact, by

M. Korkmaz, First homology group of mapping class groups of
nonorientable surfaces, Math. Proc. Cambridge Philos. Soc. 123 (1998),
487-499.

the abelianisations of all $\mathcal{N}_g$ are known: starting at $g=1$ they are
$$0, (\mathbb{Z}/2)^2, (\mathbb{Z}/2)^2, (\mathbb{Z}/2)^3, (\mathbb{Z}/2)^2, (\mathbb{Z}/2)^2, \mathbb{Z}/2, \mathbb{Z}/2, \ldots.$$

As the abelianisation does not vanish for $g > 1$ these groups are not perfect, so they do not have a universal central extension.