After just adding an SSD and an external HDD, I can no longer access File Explorer from the Metro interface.
The old C drive became the F drive and the SSD became the C drive.
The Mail app cannot add attachments, for example, and the Metro File Explorer app cannot see any folders that weren't already favourites before I made these changes.
Any ideas what's going on and what I could do?

It sounds like you have unfortunately re-assigned your old C drive by installing the SSD, which is where Windows would be installed by default. The boot order in the BIOS may have changed when you connected the SSD, but because your new SSSD does not have Windows on it, the BIOS would find the boot drive but map it differently in the BIOS causing it to become the F drive in Windows.
Try disconnecting the SSD drive and the external drive and see if it returns to normal. If it does then go into your BIOS settings and ensure that the existing C Drive is first in the boot priority after re-connecting the SSD. You may have to look up your BIOS manual on how to do this as they are all different. However, someone on here may be able to help you if you can tell us the Motherboard manufacturer, model and version - or if it is a shop bought PC, the PC manufacturer, and model.
The external drive, if connected by USB, should not be affected.