Strong atmospheric escape has been detected in several close-in exoplanets. As these planets consist mostly of hydrogen, observations in hydrogen lines, such as Lyα and Hα, are powerful diagnostics of escape. Here, we simulate the evolution of atmospheric escape of close-in giant planets and calculate their associated Lyα and Hα transits. We use a one-dimensional hydrodynamic escape model to compute physical properties of the atmosphere and a ray-tracing technique to simulate spectroscopic transits. We consider giant (0.3 and 1M jup ) planets orbiting a solar-like star at 0.045au, evolving from 10 to 5000 Myr. We find that younger giants show higher rates of escape, owing to a favourable combination of higher irradiation fluxes and weaker gravities. Less massive planets show higher escape rates (10 10 -10 13 g/s) than those more massive (10 9 -10 12 g/s) over their evolution. We estimate that the 1-M jup planet would lose at most 1% of its initial mass due to escape, while the 0.3-M jup planet, could lose up to 20%. This supports the idea that the Neptunian desert has been formed due to significant mass loss in low-gravity planets. At younger ages, we find that the mid-transit Lyα line is saturated at line centre, while Hα exhibits transit depths of at most 3 -4% in excess of their geometric transit. While at older ages, Lyα absorption is still significant (and possibly saturated for the lower mass planet), the Hα absorption nearly disappears. This is because the extended atmosphere of neutral hydrogen becomes predominantly in the ground state after ∼ 1.2 Gyr.