ABSTRACT Fast-rotating pulsars and magnetars have been suggested as the central engines of superluminous supernovae (SLSNe) and fast radio bursts, and this scenario naturally predicts non-thermal synchrotron emission from their nascent pulsar wind nebulae (PWNe). We report results of high-frequency radio observations with ALMA and NOEMA for three SLSNe (SN 2015bn, SN 2016ard, and SN 2017egm), and present a detailed theoretical model to calculate non-thermal emission from PWNe with an age of ∼1−3 yr. We find that the ALMA data disfavours a PWN model motivated by the Crab nebula for SN 2015bn and SN 2017egm, and argue that this tension can be resolved if the nebular magnetization is very high or very low. Such models can be tested by future MeV–GeV gamma-ray telescopes such as AMEGO.