Well this will only help if:
Your email provider supports imap
All your devices and computers are connecting to it through imap. If one is still in pop, You lose the benefit of it.
iMap basically synchronize your mail account between all your devices. If you read an email on your iPad, it'll be marked as read on your computer, etc... Any chance on one appears on others.
... I don't know the difference or benefit of one over the other
Also See Here Setting up and troubleshooting Mail
From Here http://www.apple.com/support/ipad/mail/
It is most likely that the Waitrose server needs to be instructed to leave a copy of emails on its server for a set number of days so that other devices can get a copy. I am not sure what email programme you use to manage your emails but here is a link to demonstrate what needs to be done if you use Outlook. Some minor adjustments will be required if it is another programme.