Hey there, and welcome to Apple Support Communities, hac66!
From my understanding, you’re not receiving Mail notifications on your iPhone when it’s locked. I’d be happy to take a look!
You mentioned that your notifications are set for banners and lock screen, but I’d recommend going to Settings > Notifications Mail, and selecting the particular account that is affected. After selecting the appropriate account, there’s an individual set of settings for allowing notifications on that account specifically. You can also check to see if Thread Notifications is enabled, so that any replies to a particular email thread will alert you as well.
I’d also suggest going to Settings > Passwords & Accounts > Fetch New Data > Select the account that is affected. From there, you’d want to determine whether Fetch, Push, or Manual is set. This will determine how your email is refreshed, and how you receive notifications.
Last but not least, I'd recommend ensuring that Do Not Disturb is turned off as well. If you see a crescent moon icon on the upper right of your iPhone, this could prevent you from receiving notifications, depending on your Settings. To check that, go to Settings > Do Not Disturb > Silence, there should be an option there for Always or While iPhone is Locked. If Do Not Disturb is off, and you don't have Scheduled enabled, this feature is also available with Bedtime alarms, if you seem to experience this when sleeping.
Let me know the results, and have a great day!
