I am specifically thinking of the Grand Tour episode entitled "A massive hunt". Is there a word that describes this wordplay? It is not a double-entendre, nor is it a homophone.
It is an extension of the classic pun
: the usually humorous use of a word in such a way as to suggest two or more of its meanings or the meaning of another word similar in sound
The most common form of puns uses multiple meanings of the same word. The second most common substitutes a homophone. And the least common uses a word which sounds similar and can include rhymes.
Examples of the third type
What do sea monsters eat? Fish and ships!
The librarian got crushed under an avalanche of books… But he only has his shelf to blame.
Why did the clown hold the door open for everyone? He made a nice jester.
Ladies, if he can’t appreciate your fruit jokes, you need to let that mango.
Some aquatic mammals at the zoo escaped. It was otter chaos!
What do you get when you mix alcohol and literature? Tequila mockingbird.
I don't think there is a particular name for puns on profanity, but you can easily describe them as dirty or naughty puns.
I suspect that, in overall context, this is a form of bathos in which the listener expects the present tone to continue in the anticipated form, only to be deceived by the mundane.
The classic example of bathos is Alexander Pope's "Rape of the Lock":
Here thou, great Anna! whom three realms obey,
Dost sometimes counsel take—and sometimes tea.
In which the dignified and formal image of Queen Anne presiding at court is destroyed by her having a cup of tea.
But it works in comedy as well:
Example #4: "I’m Sorry I’ll Read That Again" (BBC Radio Comedy)
The British radio series also provides us with many bathos examples. John Cleese and Jo Kendall appeared in the roles of a couple whose relationship is on the brink of failure.
MARY: “John – once we had something that was pure, and wonderful, and good. What’s happened to it?”
JOHN: “You spent it all.”
When Mary says “something pure and wonderful,” she is actually referring to the deep, sacred, noble form of love. However, the description is vague enough for John to manipulate.
For a tour-de-force of this form, close to your example, I suggest listening to or reading the lyrics of "The Assumption Song" by the Arrogant Worms.