Ensuring equitable physical access to SARS-CoV-2 testing has proven to be crucial for controlling the COVID-19 epidemic, especially in countries like Nepal with its challenging terrain. During the second wave of the pandemic in May 2021, there was immense pressure to expand the laboratory network in Nepal to ensure calibration of epidemic response. The expansion led to an increase in the number of testing facilities from 69 laboratories in May 2021 to 89 laboratories by November 2021. We assessed the equity of physical access to COVID-19 testing facilities in Nepal during 2021. Furthermore, we investigated the potential of mathematical optimisation in improving accessibility to COVID-19 testing facilities. Based on up-to-date publicly available data sets and on the COVID-19-related daily reports published by Nepal's Ministry of Health and Population from May 1 to November 15, 2021, we measured the disparities in geographical accessibility to COVID-19 testing across Nepal at a resolution of 1km2. In addition, we proposed an optimisation model to prescribe the best possible locations to set up testing laboratories maximizing access, and tested its potential impact in Nepal. The analysis identified vulnerable districts where, despite ramping up efforts, physical accessibility to testing facilities remains low under two modes of travel-walking and motorized driving. Both geographical accessibility and its equality were better under the motorised mode compared with the walking mode. If motorised transportation were available to everyone, the population coverage within 60min of any testing facility (public and private) would be close to threefold the coverage for pedestrians within the same hour: 61.4% motorised against 22.2% pedestrian access within the hour, considering the whole population of Nepal. Very low accessibility was found in most areas except those with private test centres concentrated in the capital city of Kathmandu. The hypothetical use of mathematical optimisation to select 20 laboratories to add to the original 69 could have improved access from the observed 61.4% offered by the laboratories operating in November to 71.4%, if those 20 could be chosen optimally from all existing healthcare facilities in Nepal. In mountainous terrain, accessibility is very low and could not be improved, even considering all existing healthcare facilities as potential testing locations. The findings related to geographical accessibility to COVID-19 testing facilities should provide valuable information for health-related planning in Nepal, especially in emergencies where data might be limited and decisions time-sensitive. The potential use of publicly available data and mathematical optimisation could be considered in the future. WHO Special Programme for Research and Training in Tropical Diseases (TDR).