Sounds like it has to do with your connection, or with your email provider's servers.I will open my email list and it will show all these new emails but take forever to load the content of the email. I won't even get a line preview for almost a minute after I have received the email.
Never had this issue before, my 5S was flawless. All started when I got the iPhone 6. I have 13 email accounts, most setup through my server for work, and the rest are gmail and yahoo accounts.Sounds like it has to do with your connection, or with your email provider's servers.
What is the provider? Do you have only one email account set up?