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Determining the habitability and interpreting atmospheric spectra of exoplanets requires understanding their atmospheric physics and chemistry. We use a 3-D Coupled Climate-Chemistry Model, the Met Office Unified Model with the UK Chemistry and Aerosols framework, to study the emergence of lightning and its chemical impact on tidally-locked Earth-like exoplanets. We simulate the atmosphere of Proxima Centauri b orbiting in the Habitable Zone of its M-dwarf star, but the results apply to similar M-dwarf orbiting planets. Our chemical network includes the Chapman ozone reactions and hydrogen oxide (HO$_{\mathrm{x}}$=H+OH+HO$_2$) and nitrogen oxide (NO$_{\mathrm{x}}$=NO+NO$_2$) catalytic cycles. We find that photochemistry driven by stellar radiation (177-850 nm) supports a global ozone layer between 20-50 km. We parameterise lightning flashes as a function of cloud-top height and the resulting production of nitric oxide (NO) from the thermal decomposition of N$_2$ and O$_2$. Rapid dayside convection over and around the substellar point results in lightning flash rates of up to 0.16 flashes km$^{-2}$yr$^{-1}$, enriching the dayside atmosphere below altitudes of 20 km in NO$_{\mathrm{x}}$. Changes in dayside ozone are determined mainly by UV irradiance and the HO$_{\mathrm{x}}$ catalytic cycle. ~45% of the planetary dayside surface remains at habitable temperatures (T$_{\mathrm{surf}}$>273.15 K) and the ozone layer reduces surface UV radiation levels to 15%. Dayside-nightside thermal gradients result in strong winds that subsequently advect NO$_{\mathrm{x}}$ towards the nightside, where the absence of photochemistry allows NO$_{\mathrm{x}}$ chemistry to involve reservoir species. Our study also emphasizes the need for accurate UV stellar spectra to understand the atmospheric chemistry of exoplanets.