  In this paper, we study a possible early universe source for the recent
observation of a stochastic gravitational wave background at the NANOGrav
pulsar timing array. The source is a tachyonic instability in a dark gauge
field induced by an axion-like particle (ALP), a known source for gravitational
waves. We find that relative to the previous analysis with the NANOGrav
12.5-year data set, the current 15-year data set favors parameter space with a
relatively larger axion mass and decay constant. This favored parameter space
is heavily constrained by $\Delta N_{\rm eff}$ and overproduction of ALP dark
matter. While there are potential mechanisms for avoiding the second problem,
evading the $\Delta N_{\rm eff}$ constraint remains highly challenging. In
particular, we find that the gravitational wave magnitude is significantly
suppressed with respect to the gauge boson dark radiation, which implies that
successfully explaining the NANOGrav observation requires a large additional
dark radiation, violating the cosmological constraints. Satisfying the $\Delta
N_{\rm eff}$ constraint will limit the potential contribution from this
mechanism to the observed signal to at most a percent level.
