I have just had a problem with iPhone mail not updating and it was a sync setting issue. To begin with, make sure that you have Background App refresh on and your mail fetch settings are set to Push or a shorter set. You should also ensure that Low Power Mode is off but this one can slow down updates. It is sometimes fixed by adding the email account again. This can also be brought about by network problems or old-fashioned iOS. I used a troubleshooting procedure on devicefixes and it assisted me in addressing the problem sequentially.