  Recent findings of anomalous super-linear 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. 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 point group symmetry reduction
result in non-universal 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: YBa$_2$Cu$_3$O$_{6.95}$ and CeCoIn$_5$. Considering all
symmetry-lowering possibilities of the point group of interest, $C_{4v}$, we
find our results to be remarkably consistent with YBa$_2$Cu$_3$O$_{6.95}$ being
near vertical nematic QCP, and CeCoIn$_5$ being near diagonal nematic QCP. Our
results motivate search for diagonal nematic fluctuations in CeCoIn$_5$.
