The "Cold Cream Guards" is a pun on the Coldstream Guards, a storied old British army regiment. The word play was also used in "The Andy Griffith Show."
The Shropshire Slasher is likely a pun on Jack the Ripper, who was never caught and never served time, let alone at HM Prison Dartmoor. Twenty or so years later, however, Britain would be plagued by a very real criminal dubbed The Yorkshire Ripper.
The sign inside of Henry the Eight's Fifth advertising " Selzer's Water" is a nod to producer Eddie Selzer.
Pudding Lane is an actual street in London EC3. It is said to have been the location of the bakery, belonging to Thomas Farriner, where a fire began that turned into the Great Fire of 1666.
Holmes and Watkins are both shown wearing Middle Eastern headgear while they are at home: a a Turkish style tarboosh, and a fez.