  The reaction $NN \to NN \pi$ offers a good testing ground for chiral
effective field theory at intermediate energies. It challenges our
understanding of the first inelastic channel in nucleon-nucleon scattering and
of the charge-symmetry breaking pattern in hadronic reactions. In our previous
studies, we presented a complete calculation of the pion-production operator
for s-wave pions up-to-and-including next-to-next-to-leading order (NNLO) in
the formulation of chiral effective field theory, which includes pions,
nucleons and $\Delta(1232)$ degrees of freedom. In this paper we calculate the
near threshold cross section for the $pp \to d \pi^{+}$ reaction by performing
the convolution of the obtained operators with nuclear wave functions based on
modern phenomenological and chiral potentials. The available chiral $NN$ wave
functions are constructed with a cutoff comparable with the momentum transfer
scale inherent in pion production reactions. Hence, a significant portion of
the dynamical intermediate-range physics is thereby cut off by them. On the
other hand, the NNLO amplitudes evaluated with phenomenological wave functions
appear to be largely independent of the $NN$ model used and give corrections to
the dominant leading order contributions as expected from dimensional analysis.
The result gives support to the counting scheme used to classify the pion
production operators, which is a precondition for a reliable investigation of
the chirally suppressed neutral pion production. The explicit inclusion of the
$\Delta(1232)$ is found to be important but smaller than expected due to
cancellations.
