The nonlinear dynamics resulting from the interplay between diffusive and buoyancy-driven Rayleigh-Taylor (RT) instabilities of autocatalytic traveling fronts are analyzed numerically for fronts ascending or descending in the gravity field and for various values of the relevant parameters, the Rayleigh numbers R(a) and R(b) of the reactant A and autocatalytic product B, respectively, and the ratio D=D(B)/D(A) of the diffusion coefficients of the two key chemical species. The interaction between the coarsening dynamics characteristic of the RT instability and the fixed short wavelength dynamics of the diffusive instability leads in some parameter regimes to complex dynamics dominated by the irregular succession of birth and death of fingers. Large single convective fingers with a tip deformed by the short wavelength diffusive instability are also observed. If D is sufficiently small and the RT instability is active, the concentration of the slower diffusing species B can be convected to values above its fully reacted concentration. Experimental conditions that would allow the observation of the dynamics predicted here are described.