I've had issues with Mail from the 13 betas and now on the released 13.2.2, not tried the 13.3 beta to see if it't any better yet.
The problem for me is that I get a banner, sound and app badge to say I've received a new email, but on opening the Mail app there is no new email showing. I have to either wait 10-20 for it to finally appear or pull down to respring the list to bring it down. It's like the background app refresh just isn't working. This is on the "All Inboxes" view of which I have iCloud, Gmail, Exchange and a couple of IMAP addresses on my device.
One thing I did notice was that in the background app refresh screen in settings Mail isn't actually listed currently, not sure if it was before if anyone knows if it should be there or not, or someone on iOS 12 can have a look?
Biggest issue i have currently however is the fact I upgraded my financee's phone to 13 a couple of weeks ago and she not stopped moaning about the mail issue since, so it's not just me!