There seems to be little information on the Internet about how to do this - the most useful article simply said “Take note of how you remove the filter so it’s not so frustrating putting the new one in”. Helpful. Especially when you (like me) removed the filter a month earlier and have not been able to find a replacement unit and have to put the old one back in temporarily.
This is what the interior cabin air filter will look like - I’ve heard it may come in a range of trim colours including white, grey and yellow. Mine is black.
The box containing the air filter is on the passenger side of the vehicle under the glovebox and has 4 clips which are quite tricky to remove - 2 at the front of the box, 2 behind. Don’t worry about putting pressure on the insulation/firewall behind the box when undoing the rear clips. My air filter box also had the Nokia car kit speaker screwed into the cover - that has since been removed.
The filter is quite long and slides on rails; Slide it out along the rails after flipping out the end closest to the door as it’s tucked in.
More information and pictures can be found through this link:
Air Conditioning Filter? - MINI2 - MINI Forum