Measurements of the second Fourier harmonic coefficient ($v_2$) of the azimuthal distributions of prompt and nonprompt D$^{0}$ mesons produced in pp and pPb collisions are presented. Nonprompt D$^{0}$ mesons come from beauty hadron decays. The data samples are collected by the CMS experiment at nucleon-nucleon center-of-mass energies of 13 and 8.16 TeV, respectively. In high multiplicity pp collisions, $v_2$ signals for prompt charm hadrons are reported for the first time, and are found to be comparable to those for light-flavor hadron species over a transverse momentum ($p_{\mathrm{T}}$) range of 2-6 GeV. Compared at similar event multiplicities, the prompt D$^{0}$ meson $v_2$ values in pp and pPb collisions are similar in magnitude. The $v_2$ values for open beauty hadrons are extracted for the first time via nonprompt D$^{0}$ mesons in D$^{0}$ collisions. For $p_{\mathrm{T}}$ in the range of 2-5 GeV, the results suggest that $v_2$ for nonprompt D$^{0}$ mesons is smaller than that for prompt D$^{0}$ mesons. These new measurements indicate a positive charm hadron $v_2$ in pp collisions and suggest a mass dependence in $v_2$ between charm and beauty hadrons in the pPb system. These results provide insights into the origin of heavy-flavor quark collectivity in small systems.