This paper reports the measurement of $\mathrm{J}/{\psi }$ meson production in proton–proton ( $\mathrm {p}\mathrm {p}$ ) and proton–lead ( $\mathrm {p}\mathrm {Pb}$ ) collisions at a center-of-mass energy per nucleon pair of $5.02,\text {TeV} $ by the CMS experiment at the LHC. The data samples used in the analysis correspond to integrated luminosities of 28 $,\text {pb}^{-1}$ and 35 $,\text {nb}^{-1}$ for $\mathrm {p}\mathrm {p}$ and $\mathrm {p}\mathrm {Pb}$ collisions, respectively. Prompt and nonprompt $\mathrm{J}/{\psi }$ mesons, the latter produced in the decay of ${\mathrm {B}}$ hadrons, are measured in their dimuon decay channels. Differential cross sections are measured in the transverse momentum range of $2<p_{\mathrm {T}} <30{,\text {GeV}/{c}} $ , and center-of-mass rapidity ranges of $|y_\mathrm{{CM}} |<2.4$ ( $\mathrm {p}\mathrm {p}$ ) and $-2.87<y_\mathrm{{CM}}<1.93$ ( $\mathrm {p}\mathrm {Pb}$ ). The nuclear modification factor, $R_{\mathrm {p}\mathrm {Pb}}$ , is measured as a function of both $p_{\mathrm {T}}$ and $y_\mathrm{{CM}}$ . Small modifications to the $\mathrm{J}/{\psi }$ cross sections are observed in $\mathrm {p}\mathrm {Pb}$ relative to $\mathrm {p}\mathrm {p}$ collisions. The ratio of $\mathrm{J}/{\psi }$ production cross sections in $\mathrm {p}$ -going and Pb-going directions, $R_\mathrm{{FB}}$ , studied as functions of $p_{\mathrm {T}}$ and $y_\mathrm{{CM}}$ , shows a significant decrease for increasing transverse energy deposited at large pseudorapidities. These results, which cover a wide kinematic range, provide new insight on the role of cold nuclear matter effects on prompt and nonprompt $\mathrm{J}/{\psi }$ production.