  The detection of GeV $\gamma$-ray emission from Galactic novae by $Fermi$-LAT
has become routine since 2010, and is generally associated with shocks internal
to the nova ejecta. These shocks are also expected to heat plasma to $\sim
10^7$ K, resulting in detectable X-ray emission. In this paper, we investigate
13 $\gamma$-ray emitting novae observed with the Neil Gehrels $Swift$
Observatory, searching for 1-10 keV X-ray emission concurrent with $\gamma$-ray
detections. We also analyze $\gamma$-ray observations of novae V407 Lup (2016)
and V357 Mus (2018). We find that most novae do eventually show X-ray evidence
of hot shocked plasma, but not until the $\gamma$-rays have faded below
detectability. We suggest that the delayed rise of the X-ray emission is due to
large absorbing columns and/or X-ray suppression by corrugated shock fronts.
The only nova in our sample with a concurrent X-ray/$\gamma$-ray detection is
also the only embedded nova (V407 Cyg). This exception supports a scenario
where novae with giant companions produce shocks with external circumbinary
material and are characterized by lower density environments, in comparison
with novae with dwarf companions where shocks occur internal to the dense
ejecta.
