Young massive clusters (YMCs) are the most massive star clusters forming in nearby galaxies and are thought to be a young analogue to the globular clusters. We study possible observational signatures of their formation stage, particularly when the mass of a cluster is increasing via accretion from a natal molecular cloud. To this end, we study the broad-band continuum emission from ionized gas and dust near YMCs, whose formation is followed by recent radiation-hydrodynamics simulations. We perform post-process radiative transfer calculations using simulation snapshots and find characteristic spectral features at radio and far-infrared wavelengths. We show that a striking feature is long-lasting, strong free-free emission from a ~10pc-scale HII region. Because of its large emission measure, there is a turnover feature below ~10GHz, a signature of the free-free absorption, often found in Galactic ultra-compact HII regions. These features come from the peculiar YMC formation process, where the cluster's gravity effectively traps photoionized gas for a long duration and enables continuous star formation within the cluster. Such large and dense HII regions show distinct distribution on the density-size diagram, apart from the standard sequence of Galactic HII regions. This is consistent with the observational trend inferred for extragalactic HII regions associated with YMCs.