HD :
There were nightly BM mail flights on Heathrow to Edinburgh, they were non-pax and departed each end at about 23.30. They provided one extra sector at the end of the day for the two LHR-EDI aircraft. They were actually charters by Royal Mail and had flight numbers in the BMA charter flight series, not regular numbers for the Edinburgh service. I'm not certain if the night landing slot came from a Post Office allocation rather than a BMI one.
The aircraft were not convertible as such for the mail run but there were custom-made kits that fitted over the seats that the mail pouches were put into, it took about half an hour to set the cabin up. BMA operated this for quite some years until a Post Office reorganisation ended the service from Heathrow.
Regarding the landing procedure, others are doubtless able to comment.