In a reference system moving with the bubble vertex we investigate the effects of fluid viscosity and surface tension on the bubble velocity in the nonlinear Rayleigh-Taylor (RT) and Richtmyer-Meshkov (RM) instabilities, by extending the ideal fluid model [Goncharov V N, Phys. Rev. Lett. 88 134502 (2002)] to the non-ideal fluid case. First of all, the governing equation (i.e. self-consistent differential equations) describing the dynamic of the bubble front in RT and RM instabilities is obtained. Then, the numerical and asymptotic solutions of the bubble velocity in two-dimensional planar geometry and three-dimensional cylindrical geometry are obtained. Moreover, we quantitatively study the effects of fluid viscosity and surface tension on the RT and RM bubble velocities. It is found that in the fully nonlinear evolutions of RT and RM instabilities, the bubble velocity and amplitude in the non-ideal fluid are both less than those in its ideal fluid counterpart. That is to say, the effects of fluid viscosity and surface tension tend to stabilize the RT and RM instabilities.