Go to Settings –> Cellular, scrolled down on “Use Cellular Data For:” And here’s how it looked (this is iOS 10– other versions may look a little different).
Scrolling down a little more until seeing this:
Check that “Use Cellular Data For” was turned OFF or not for Mail! So many apps were allowed, but not Mail. Naturally I flipped the switch for Mail to “On” and the problem went away.
Note: if, for some reason, you turn “Use Cellular Data For” to OFF for Mail, you’ll get a big warning telling you your mail won’t work right if you do that. Apparently, with User’s iPhone, the warning was not shown, or not seen, or not understood– certainly not remembered.
UPDATE: in iOS 13, Mail’s settings include the Use Cellular Data switch. If you turn it on here, it turns it on in the “Cellular Data/Use Cellular Data For” section. One setting, in two places. Looks like this:
Summing it up
As usual, solving this problem started with figuring out what the problem was. We started out thinking the problem involved the iPhone settings for User’s email accounts, but that wasn’t it. Then we thought cellular data might be off– but we saw it was on. Then we thought it might be something wrong with the cellular antenna, but we were able to use Safari over cellular, so that wasn’t it either. Then we thought there might be something wrong with user’s account with the cellular carrier– but that wasn’t it (TELUS said so, and again we knew we could use Safari, and that uses the same cellular data as the Mail app).
Finally, having looked at the master switch for Cellular data a second time, we had the idea of scrolling down (on a Settings page that doesn’t look as if it has anything to scroll). I think it was just bad luck that the Cellular settings screen fit so perfectly on the iPhone 7– had it been cut off in the middle or something, we probably would have realized that there were more settings below and that we should scroll down to see them. Anyhow, once we saw the “Use Cellular Data For:” section, we zeroed in on the problem, and fixed it right away.
Ref: https://christianboyce.com/iphone-no-mail-with-cellular-data/