Atmospheric trace gas measurements can be used to independently assess national greenhouse gas inventories through inverse modelling. Here, atmospheric nitrous oxide (N2O) measurements made in the United Kingdom (U.K.) and Republic of Ireland are used to derive monthly N2O emissions for 2013-2022 using two different inverse methods. We find mean U.K. emissions of 90.5±23.0 (1\(\sigma\)) and 111.7±32.1 (1\(\sigma\)) Gg N2O yr-1 for 2013-2022, and corresponding trends of -0.68±0.48 (1\(\sigma\)) Gg N2O yr-2 and -2.10±0.72 (1\(\sigma\)) Gg N2O yr-2, respectively for the two inverse methods. The U.K. National Atmospheric Emissions Inventory (NAEI) reported mean N2O emissions of 73.9 Gg N2O yr-1 across this period, which is 14-33% smaller than the emissions derived from atmospheric data. We infer a pronounced seasonal cycle in N2O emissions, with a peak occurring in the spring and a second smaller peak in the late summer for certain years. The springtime peak has a long seasonal decline that contrasts with the sharp rise and fall of N2O emissions estimated from the bottom-up U.K. Emissions Model (UKEM). Bayesian inference is used to minimize the seasonal cycle mismatch between the average top-down (atmospheric data-based) and bottom-up (process model and inventory-based) seasonal emissions at a sub-sector level. Increasing agricultural manure management and decreasing synthetic fertilizer N2O emissions reduces some of the discrepancy between the average top-down and bottom-up seasonal cycles. Other possibilities could also explain these discrepancies, such as missing emissions from NH3 deposition, but these require further investigation.