The quasiparticle spectra of atomically thin semiconducting transition metal dichalcogenides (TMDCs) and their response to an ultrafast optical excitation critically depend on interactions with the underlying substrate. Here, we present a comparative time- and angle-resolved photoemission spectroscopy (TR-ARPES) study of the transient electronic structure and ultrafast carrier dynamics in the single- and bilayer TMDCs MoS$_2$ and WS$_2$ on three different substrates: Au(111), Ag(111) and graphene/SiC. The photoexcited quasiparticle bandgaps are observed to vary over the range of 1.9-2.3 eV between our systems. The transient conduction band signals decay on a sub-100 fs timescale on the metals, signifying an efficient removal of photoinduced carriers into the bulk metallic states. On graphene, we instead observe two timescales on the order of 200 fs and 50 ps, respectively, for the conduction band decay in MoS$_2$. These multiple timescales are explained by Auger recombination involving MoS$_2$ and in-gap defect states. In bilayer TMDCs on metals we observe a complex redistribution of excited holes along the valence band that is substantially affected by interactions with the continuum of bulk metallic states.