Here's a historical reenactment starring US Army veteran and historical interpreter Tim Conway, demonstrating the value of shot tongs and why they were necessary...
While normally associated with mortars, these tongs were also use on the 12.75" Armstrong gun in Charleston and other extremely large bore guns along with a loading derrick.
The only two terms that I could come across during my search were "Cannonball Cradle" and "Cannon Rammer". They were most likely just called "tongs" by that used them. I'm not really sure what they are actually called.
Here's a historical reenactment starring US Army veteran and historical interpreter Tim Conway, demonstrating the value of shot tongs and why they were necessary...