The ion-implantation-perturbed-angular-correlation technique has been used to measure the Larmor precessions of the first ${2}^{+}$ excited states of $^{182,184,186}\mathrm{W}$ in a ferromagnetic nickel host. The observed angular precessions, corrected for the transient field effect, were $\ensuremath{\omega}{\ensuremath{\tau}}_{182}({2}^{+})=0.159\ifmmode\pm\else\textpm\fi{}0.003$ rad, $\ensuremath{\omega}{\ensuremath{\tau}}_{184}({2}^{+})=0.156\ifmmode\pm\else\textpm\fi{}0.011$ rad, and $\ensuremath{\omega}{\ensuremath{\tau}}_{186}({2}^{+})=0.160\ifmmode\pm\else\textpm\fi{}0.007$ rad. Use is made of the accepted $g$ factor of $^{184}\mathrm{W}$, ${g}_{184}({2}^{+})=0.288\ifmmode\pm\else\textpm\fi{}0.007$, to obtain ${H}_{\mathrm{hf}}(Ni\ensuremath{-}\mathrm{W})=(\ensuremath{-}67\ifmmode\pm\else\textpm\fi{}5)$ kOe at 305 K. The $g$ factors of $^{182,186}\mathrm{W}$, determined relative to the known $g$ factor of $^{184}\mathrm{W}$, are ${g}_{182}({2}^{+})=0.271\ifmmode\pm\else\textpm\fi{}0.021$, and ${g}_{186}({2}^{+})=0.370\ifmmode\pm\else\textpm\fi{}0.036$. Larmor precessions were also measured for the first ${2}^{+}$ excited states of $^{184,186}\mathrm{W}$ in a ferromagnetic cobalt host. The corrected observed angular precessions were $\ensuremath{\omega}{\ensuremath{\tau}}_{184}({2}^{+})=0.839\ifmmode\pm\else\textpm\fi{}0.019$ and $\ensuremath{\omega}{\ensuremath{\tau}}_{186}({2}^{+})=0.833\ifmmode\pm\else\textpm\fi{}0.015$. The resulting hyperfine field on tungsten in cobalt, assuming ${g}_{184}({2}^{+})=0.288\ifmmode\pm\else\textpm\fi{}0.007$, is ${H}_{\mathrm{hf}}(Co\ensuremath{-}\mathrm{W})=(\ensuremath{-}341\ifmmode\pm\else\textpm\fi{}12)$ kOe, and the resulting $g$ factor of the first excited state of $^{186}\mathrm{W}$, determined relative to ${g}_{184}({2}^{+})$ in the cobalt host, is ${g}_{186}({2}^{+})=0.358\ifmmode\pm\else\textpm\fi{}0.021$. Agreement between ${g}_{186}({2}^{+})$ obtained from using a nickel host and ${g}_{186}({2}^{+})$ obtained using a cobalt host was very good, and a weighted average of these two independent measurements gives ${g}_{186}({2}^{+})=0.361\ifmmode\pm\else\textpm\fi{}0.018$. Target temperature measurements as a function of beam current were also performed. The observed trend of $g$ with increasing $A$ is not compatible with present theories.[NUCLEAR REACTIONS $^{182,184,186}\mathrm{W}(^{16}\mathrm{O},^{16}\mathrm{O}^{\ensuremath{'}}\ensuremath{\gamma})$, $E=36$ MeV; measured $^{16}\mathrm{O}\ensuremath{\gamma}$-coin, $\ensuremath{\gamma}\ensuremath{\gamma}(\ensuremath{\theta},\mathrm{H})$ in Ni, Co; deduced hyperfine fields. NUCLEAR MOMENTS $^{182,184,186}\mathrm{W}$; deduced $g$ for 100, 111, 123 keV levels.]