Recent findings of anomalous superlinear scaling of low-temperature (T) penetration depth (PD) in several nodal superconductors near putative quantum critical points suggest that the low-temperature PD can be a useful probe of quantum critical fluctuations in a superconductor. On the other hand, cuprates, which are poster child nodal superconductors, have not shown any such anomalous scaling of PD, despite growing evidence of quantum critical points (QCP). Then it is natural to ask when and how can quantum critical fluctuations cause anomalous scaling of PD? Carrying out the renormalization group calculation for the problem of two-dimensional superconductors with point nodes, we show that quantum critical fluctuations associated with a point group symmetry reduction result in nonuniversal logarithmic corrections to the T dependence of the PD. The resulting apparent power law depends on the bare velocity anisotropy ratio. We then compare our results to data sets from two distinct nodal superconductors: YBa2Cu3O6.95 and CeCoIn5. Considering all symmetry-lowering possibilities of the point group of interest, C4v, we find our results to be remarkably consistent with YBa2Cu3O6.95 being near a vertical nematic QCP and CeCoIn5 being near a diagonal nematic QCP. Our results motivate a search for diagonal nematic fluctuations in CeCoIn5. ©2015 American Physical Society.