THIS PROBLEM HAS NOW BEEN FIXED BY APPLE!

This article will be of interest to you if:

One of your email ISPs uses POP3 e.g. paradise.net.nz, clear.net
You have an iPad or iPhone, running software updated to IOS 9 or 9.1
You read/collect emails on the device
You have suffered the problem of disappearing email.

Here’s what happens:

you received an email from the POP3 ISP.
You opened this email on your iOS9 device and read it.
At a later time, you went back to that email, with the idea of re-reading it, or maybe even replying to it.
You found the email in your Inbox as before. You could see who sent it, and when, the subject line, and the first few words of the email.
You tapped on it, expecting it to open. It did, but instead of the content of the email you found the words
This message has not been downloaded from the server.

This “disappearing email content phenomenon” is a real puzzle for Apple, because it doesn’t happen to everyone using a POP3 ISP, nor is every email affected. It has only been occurring since the update to IOS9. Apple are working on it, and will presumably fix the problem in the near future, and we can turn iCloud/Mail back on

Until Apple does fix this problem, there are a couple of things you can do to ensure you keep the content of your POP3 emails until you are ready to delete them.

Use your browser (e.g Safari) to go into the website of your ISP (you will need your email password!!). There you should find something akin to Settings, in which you can instruct the ISP not to delete downloaded (or read) emails. All emails which arrive thereafter while still downloading to your device, will also be kept on the ISP server, and can be read and replied to as normal. The downside of this workaround is that your Inbox at the ISP server will fill up, and when it does, you’ll get a message telling you that your “Inbox is full”. At that point you will need to log into the ISP Mail system again and manually delete all the unwanted messages (sorry!). Lets hope that before that time Apple has fixed the problem and you can reset the “do not delete” switch back to “delete downloaded messages”.
On your Apple device, go into Settings/iCloud, and turn OFF Mail. This will have the effect of knocking out email from the iCloud.com address, if you use this mail system too. But it can still be received on any other Apple device that isn’t also trying to access the POP3 Mail on which you receive mail from your iCloud.com address.
Once Mail is turned Off in Settings/iCloud, emails from a POP3 ISP will go straight to your device, and stay there, in full, until you delete them. It will pay to make sure that iCloud Drive is not on. (Setting/iCloud/iCloud Drive), and for the time being, have Backup off as well. Result: no further mail involvement with iCloud.
Note: eMails from IMAP servers like Gmail or Outlook.com seem to be unaffected.

This is a very specific problem and you should not apply the workaround above unless you have actually experienced the issue. Apple have told us that they are working to resolve it, and they will presumably fix the problem in the near future, (and we can turn iCloud/Mail back on)

In the mean time, do your Backups to iTunes just to be sure.