Aims. We study the process of magnetic field annihilation and reconnection in simulations of magnetised solar photosphere and chromosphere with magnetic fields of opposite polarities and constant numerical resistivity. Methods. Exact analytical solutions for reconnective annihilations were used to interpret the features of magnetic reconnection in simulations of flux cancellation in the solar atmosphere. We used MURaM high-resolution photospheric radiative magneto-convection simulations to demonstrate the presence of magnetic field reconnection consistent with the magnetic flux pile-up models. Also, a simulated data-driven chromospheric magneto-hydrodynamic simulation is used to demonstrate magnetic field and flow structures, which are similar to the theoretically predicted ones. Results. Both simulations demonstrate flow and magnetic field structures roughly consistent with accelerated reconnection with magnetic flux pile-up. The presence of standard Sweet-Parker type reconnection is also demonstrated in stronger photospheric magnetic fields.