SAM's e-mail notices in fact mean you have strike teams idling. It's a bug introduced with patch 1.05. No reason to check for e-mails before you finish a priority or allies mission.
Loyalty mission progress depends on different missions you completed (not tasks, these do not affect it). If it's currently ON HOLD, do some other mission, it'll unlock eventually. That goes both for Peebee's remnant scanner and Cora's whichever step.
Finally, if you didn't receive any e-mail and a mission is not on hold in journal anymore, go talk to your crewmember.
Thank you for the helpful info. I was trying to clear all side quests on Eos before doing more priority missions hence the loyalty quests are indeed "on hold". I have no issue with going back to Eos after more missions just didn't want to miss anything.
