What size nut does it take to fit the rear wheel with a nexus Shimano rear axle? I measured the axle size and I'm getting 9.2mm. or .360 in.
Most likely you need: 3/8" diameter 26 tpi (thread per inch)
On the website below there is a list off all Shimano gear hubs which use this size nut: https://www.sjscycles.co.uk/hub-spares/shimano-nexus-brim40-cap-nut-38-inch/
Please check if your hub is listed to ensure compatibility.