OK – Yes, we had the Default Order Status set to a custom status, and I had changed it thinking that might have been the cause. However, I had specified Processing (which was the WooCommerce default) there, as I either didn’t notice or didn’t understand the No Changes option at that time. After getting your response I did switch this to No Changes, which resolved the issue.
Thank you very much for your help with this! Much appreciated.
No problem! Just standard WooCommerce mail. No other SMTP plugins or anything like that in place.