The dynamics of $A+B\rightarrow C$ fronts is analysed numerically in a radial geometry. We are interested to understand miscible fingering instabilities when the simple chemical reaction changes the viscosity of the fluid locally and a non-monotonic viscosity profile with a global maximum or minimum is formed. We consider viscosity-matched reactants $A$ and $B$ generating a product $C$ having different viscosity than the reactants. Depending on the effect of $C$ on the viscosity relative to the reactants, different viscous fingering (VF) patterns are captured which are in good qualitative agreement with the existing radial experiments. We have found that, for a given chemical reaction rate, an unfavourable viscosity contrast is not always sufficient to trigger the instability. For every fixed Péclet number ($Pe$), these effects of chemical reaction on VF are summarized in the Damköhler number ($Da$) $-$ the log-mobility ratio ($R_{c}$) parameter space that exhibits a stable region separating two unstable regions corresponding to the cases of more and less viscous product. Fixing $Pe$, we determine $Da$-dependent critical log-mobility ratios $R_{c}^{+}$ and $R_{c}^{-}$ such that no VF is observable whenever $R_{c}^{-}\leqslant R_{c}\leqslant R_{c}^{+}$. The effect of geometry is observable on the onset of instability, where we obtain significant differences from existing results in the rectilinear geometry.