iPhone 8 Plus emails not coming automatically

If you messages are not loading until you open the mail app, then that sounds like it is a problem with fetching your mail, not a notification issue.

If you go to Settings>Accounts & Passwords>Fetch New Data, then select the account in question, what is the setting there? Not all accounts support Push, most only support Fetch. Then you also need to have the time setting selected for the time period that you wish to have your mail account fetch your mail.

It is also possible there is a problem with your particular account. Who is the mail provider?

It may be my misunderstanding of what Automatically means when you have that set in Fetch, but I've never seen any result from setting it there. I have mine set to 15 minutes, or you can set something else. I would test one of the other settings to see if they work. I've tried locating some information about Automatically in the Fetch setting, and I've never found any.

I found an explanation on a Verizon link for the iPhone, https://www.verizonwireless.com/support/knowledge-base-206873/, and it states that Automatically means that it Fetches in the background only when plugged into power and on wi-fi.

If you are not on power, I would guess then that you need to have a time setting for Fetch.

You're welcome. While I saw that you said you were on wi-fi, you did not indicate that you had the device plugged into power. I'll watch to see if you are successful with the time setting. As I said earlier, this is the only explanation I've seen, and I've never used that setting before on my devices. I've always used time, that way it doesn't matter if I'm on wi-fi or cellular. My mail is always fetched at the same time, without interruption.

Not a problem. I changed the setting on my one account from Fetch 15 minutes and set it to Automatically. I power cycled the phone, connected it to power and tried sending myself several test emails from another account. Only got them after I opened mail I waited over 30 minutes. Changed it back to the Fetch 15 I originally had it on. Glad you are satisfied with the outcome of this as well.