Aims. We aim to investigate the onset of 2D time-dependent magnetic reconnection that is triggered using an external (non-local) velocity driver located away from, and perpendicular to, an equilibrium Harris current sheet. Previous studies have typically utilised an internal trigger to initiate reconnection, for example initial conditions centred on the current sheet. Here, an external driver allows for a more naturalistic trigger as well as the study of the earlier stages of the reconnection start-up process. Methods. Numerical simulations solving the compressible, resistive magnetohydrodynamic (MHD) equations were performed to investigate the reconnection onset within different atmospheric layers of the Sun, namely the corona, chromosphere and photosphere. Results. A reconnecting state is reached for all atmospheric heights considered, with the dominant physics being highly dependent on atmospheric conditions. The coronal case achieves a sharp rise in electric field (indicative of reconnection) for a range of velocity drivers. For the chromosphere, we find a larger velocity amplitude is required to trigger reconnection (compared to the corona). For the photospheric environment, the electric field is highly dependent on the inflow speed; a sharp increase in electric field is obtained only as the velocity entering the reconnection region approaches the Alfvén speed. Additionally, the role of ambipolar diffusion is investigated for the chromospheric case and we find that the ambipolar diffusion alters the structure of the current density in the inflow region. Conclusions. The rate at which flux enters the reconnection region is controlled by the inflow velocity. This determines all aspects of the reconnection start-up process, that is, the early onset of reconnection is dominated by the advection term in Ohm’s law in all atmospheric layers. A lower plasma-β enhances reconnection and creates a large change in the electric field. A high plasma-β hinders the reconnection, yielding a sharp rise in the electric field only when the velocity flowing into the reconnection region approaches the local Alfvén speed.