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We present optical, radio, and X-ray observations of SN2020bvc (=ASASSN20bs; ZTF20aalxlis), a nearby ($z=0.0252$; $d$=114 Mpc) broad-lined (BL) Type Ic supernova (SN). Our observations show that SN2020bvc shares several properties in common with the Ic-BL SN2006aj, which was associated with the low-luminosity gamma-ray burst (LLGRB) 060218. First, the 10 GHz radio light curve is on the faint end of LLGRB-SNe ($L_\mathrm{radio} \approx 10^{37}$erg/s): we model our VLA observations (spanning 13-43 d) as synchrotron emission from a mildly relativistic ($v \gtrsim 0.3c$) forward shock. Second, with Swift and Chandra we detect X-ray emission ($L_X \approx 10^{41}$erg/s) that is not naturally explained as inverse Compton emission or as part of the same synchrotron spectrum as the radio emission. Third, high-cadence ($6\times$/night) data from the Zwicky Transient Facility (ZTF) shows a double-peaked optical light curve, the first peak from shock-cooling emission from extended low-mass material (mass $M<10^{-2} M_\odot$ at radius $R>10^{12}$cm) and the second peak from the radioactive decay of Ni-56. SN2020bvc is the first confirmed double-peaked Ic-BL SN discovered without a GRB trigger, and shows X-ray and radio emission similar to LLGRB-SNe: this is consistent with models in which the same mechanism produces both the LLGRB and the shock-cooling emission. For four of the five other nearby ($z\lesssim0.05$) Ic-BL SNe with ZTF high-cadence data, we rule out a first peak like that seen in SN2006aj and SN2020bvc, i.e. that lasts $\approx 1$d and reaches a peak luminosity $M \approx -18$. X-ray and radio follow-up observations of future such events will establish whether double-peaked optical light curves are indeed predictive of LLGRB-like X-ray and radio emission.