Mud Hut is on the right track. I'll speak up to confirm, it is definitely a Livery button. In this case, it represents two houses of British Nobility. It's very unusual for a Livery button to show two family Crests... so my guess is, this button commemorates the unification of the houses of two Nobles, through a marriage.
The key ID-clue for this button is the presence of a Heraldry device known as a Torse, located underneath the Family Crest. A Torse is a wreath, but viewed "on edge" (as if it was laying flat on the ground below the Crest), instead of the usual standing-up view of a wreath. A Torse always has 6 furls. It is clearly visible on the two Livery buttons in the images below. (Click on the image to enlarge it for a better view of the Torse and its 6 "furls."
Here's a link to a wonderful website showing the specific Nobility-family identification of hundreds of Livery buttons. Fyrffytr1, you can research the details of the two crests on your button to identify them.
After-posting edit:
Since you are a Friend, and also a member of the Uniformed Public Services (Military, Police, Firefighters), I researched the emblems for you. The porcupine is here:
Porcupine - Livery Buttons Identified (google.com) Family is Fyler or Prestwich and Prestwick.
There is no entry at Silversimon's Livery-Buttons site for an alligator or a crocodile. So, I suggest you try to decipher the motto in the ribbon above the gator/croc's head, and search for the motto's Livery connection.
Livery Buttons Identified (google.com)