  Atmospheric escape in exoplanets has traditionally been observed using
hydrogen Lyman-$\alpha$ and H-$\alpha$ transmission spectroscopy, but more
recent detections have utilised the metastable helium triplet at 1083$~$nm.
Since this feature is accessible from the ground, it offers new possibilities
for studying atmospheric escape. Our goal is to understand how the
observability of escaping helium evolves during the lifetime of a highly
irradiated gas giant. We extend our previous work on 1-D self-consistent
hydrodynamic escape from hydrogen-only atmospheres as a function of planetary
evolution to the first evolution-focused study of escaping hydrogen-helium
atmospheres. Additionally, using these novel models we perform helium triplet
transmission spectroscopy. We adapt our previous hydrodynamic escape model to
now account for both hydrogen and helium heating and cooling processes and
simultaneously solve for the population of helium in the triplet state. To
account for the planetary evolution, we utilise evolving predictions of
planetary radii for a close-in 0.3$~M_{\rm Jup}$ gas giant and its received
stellar flux in X-ray, hard and soft EUV, and mid-UV wavelength bins assuming a
K dwarf stellar host. We find that the helium triplet signature diminishes with
evolution. Our models suggest that young ($\lesssim 150$~Myr), close-in gas
giants ($\sim 1$ to $2~R_{\rm Jup}$) should produce helium 1083$~$nm transit
absorptions of $\sim 4\%$ or $\sim 7\%$, for a slow or fast-rotating K dwarf,
respectively, assuming a 2$\%$ helium abundance.
