"If you can it sounds like your email account has been downgraded to a BTMail Basic account. This can only be used with a web browser and not using an email app such as you will have on your phone."
But the OP said that their phone immediately picks up emails when it hooks onto his house WiFi, so the email app on his phone is picking up his emails OH then. It just doesn't when connected via 4G, and I think the OPs later suggestion that it is just too slow to pick them up which is the problem.
4G speed (and indeed 5G speed) can be very variable, depending on the strength of the signal
