In recent years, several young massive star clusters (YMSCs) have been associated with extended $\gamma$-ray sources, suggesting that some acceleration process, able to accelerate particles at least up to hundreds of TeV, is at work. The number of YMSCs with associated $\gamma$-ray emission is of order ten, however, the number of potential sources is much larger, probably up to several hundreds. It is plausible that many of such objects have not been observed yet due to their low surface brightness. However, such unresolved sources may contribute to the diffuse Galactic $\gamma$-ray background. In this work, we aim at estimating the total contribution of unresolved YMSCs to the diffuse $\gamma$-ray flux, considering a synthetic population built from observed properties of local (within ~2 kpc from the Sun) clusters. We simulate the Galactic population of YMSCs using a Montecarlo approach. For every cluster, we build the stellar population, in order to estimate the collective wind luminosity and mass loss rate. The $\gamma$-ray emission of each cluster is then computed assuming a pure hadronic scenario, where protons are accelerated at the collective wind termination shock of the stellar cluster and subsequently interact with the material embedded inside the wind-blown bubble. We also account for different scenarios for the particle diffusion inside the bubble, which determines both the maximum energy and the escape time from the bubble, thus affecting the final $\gamma$-ray spectrum. The results are then compared with measurements of the diffuse $\gamma$-ray flux provided by several experiments, from a few GeV to hundreds of TeV.