  The NANOGrav Collaboration reported strong Bayesian evidence for a
common-spectrum stochastic process in its 12.5-yr pulsar timing array dataset,
with median characteristic strain amplitude at periods of a year of $A_{\rm yr}
= 1.92^{+0.75}_{-0.55} \times 10^{-15}$. However, evidence for the quadrupolar
Hellings \& Downs interpulsar correlations, which are characteristic of
gravitational wave signals, was not yet significant. We emulate and extend the
NANOGrav dataset, injecting a wide range of stochastic gravitational wave
background (GWB) signals that encompass a variety of amplitudes and spectral
shapes, and quantify three key milestones: (I) Given the amplitude measured in
the 12.5 yr analysis and assuming this signal is a GWB, we expect to accumulate
robust evidence of an interpulsar-correlated GWB signal with 15--17 yrs of
data, i.e., an additional 2--5 yrs from the 12.5 yr dataset; (II) At the
initial detection, we expect a fractional uncertainty of $40\%$ on the
power-law strain spectrum slope, which is sufficient to distinguish a GWB of
supermassive black-hole binary origin from some models predicting more exotic
origins;(III) Similarly, the measured GWB amplitude will have an uncertainty of
$44\%$ upon initial detection, allowing us to arbitrate between some population
models of supermassive black-hole binaries. In addition, power-law models are
distinguishable from those having low-frequency spectral turnovers once 20~yrs
of data are reached. Even though our study is based on the NANOGrav data, we
also derive relations that allow for a generalization to other pulsar-timing
array datasets. Most notably, by combining the data of individual arrays into
the International Pulsar Timing Array, all of these milestones can be reached
significantly earlier.
