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.