I could say diaper-changing but could also say changing diapers. This answer will be complex: in short, the singular tends to refer to a single instance of changing a diaper or forms most compounds involving diaper-related items; the plural tends to refer to a state of wearing diapers or a repetitive action of changing diapers. There is some crossover, or specific uses where the plural diapers would be used in place of the singular, but those uses are fairly restricted.
A little bit of history: diaper originally referred to a pattern of cloth that involves crossing, especially in a diamond or diamond-like shape. This sense transferred to the cloths (towels and napkins) that bore the pattern, and then to one of the uses of said cloths: cleaning up after a baby. By the 19th century the diaper's modern use as soilable underwear for babies was widespread. (In the UK and related countries nappy formed to refer to the same garment.)
As a word, diaper has several uses in the singular. It forms several compounds, like diaper bag (where one stores diapers during travel), diaper pail (where dirty diapers are disposed), diaper rash (a rash commonly associated with diaper use), and diaper cake (a cake-shaped arrangement of many diapers made for baby showers). (All are attested by the Oxford English Dictionary.)
The search results for "change a diaper" suggest the usage is very common: the first results are for advice on how to change a diaper.
Can diapers function in the same way as pants? This Language Log post suggests it sometimes can, although the use of "diapers" in the plural to refer to a single diaper is limited to some cases where the dual sense bleeds over from similar garments. (The post and its follow-ups are all worth a read, since precisely how similar to pants the word is remains contentious.)
What about other uses of diapers in the plural?
"Change diapers" appears to be used to describe the activity in general or the activity repeated over time, like this:
I change diapers fairly often.
That said, I can think of at least one meaning works with diapers but not diaper: when someone's literal or metaphorical state (suggesting inexperience) involves being in a diaper, we can say they are in diapers, describing a state and acting as a metonym for early childhood. This usage is a bit subtle for a Google search but emerges in a Corpus of Contemporary American English search. For instance, here's Bellamy Young in a 2017 interview with Fox News describing her character "Mellie" from ABC's show Scandal:
" You know, Mellie was president in her own mind she since she was in diapers so I think she concerns herself less -- it's more of an annoyance for her to be noticed for what she's wearing or how her looks or whatever, " said Young. # " In her heart, she knows her policy, she knows her politics, she has a vision of what she wants to achieve for the country, " she added.
The multifaceted or rambling nature of this answer should suggest how varied the uses of diaper and diapers are.