I'm renovating an old property and am updating the covers on the light switches.

When removing the cover of an old switch (having turned off the mains etc) a 4 mm piece of wire broke off the red wire. There's no slack in the wire at all and now neither the old switch cover nor the new one will fit into the housing. The cover sits about 4mm above the housing and of course the screws won't line up to secure the cover to the wall.

Any suggestions?

Yep, pigtail a piece of starnded wire with a wirenut.