Hello Lewis MacLennan,
Thanks for your post in Microsoft Community. Regarding your description about “Outlook Notification Centre - Emails won't open”, may I know are you able to receive the
pop-up notification as per the below screenshot? If yes, you can open this email by clicking the pop-up window.
· 
If you cannot see
the pop-up window
for new incoming
emails, I would like to suggest you to check the below settings for new message alert pop-up notification is enable or not.
· Steps to Turn new message alert pop-up on or off:
· Open Outlook client -> File Tab-> Options-> Mail-> Message arrival -> make sure this settings is enable
· 
Article for Turn new message alert pop-up on or off: https://support.office.com/en-us/article/turn-new-message-alert-pop-up-on-or-off-9940c70e-b306-442e-a856-d94b20318481
Further, could you please kindly provide us
below information?
· May I know how you would open this email? By clicking on
the message icon in taskbar?
· Could you please kindly provide us the detailed information of your Outlook version?
Thanks and regards,
Ankita Vaidya