It's not atypical that the same address can be both affectionate when used in friendly jest, and derogatory when used in malice ("shrimp", from real life, or "seaweed brain" from Percy Jackson novels).
Wormtail clearly falls into the same cohort - it may be an affectionate (to an extent) nickname when used by Marauders but works equally well as an expression of contempt for a weak worthless thing that DE's saw Peter as.
Here's the quote illustrating it from Percy Jackson:
But Thalia held out her spear.
“You want some, Seaweed Brain?”
Somehow, it was okay when Annabeth called me that— at least, I’d gotten used to it—but hearing it from Thalia was not cool.
scifi.stackexchange.com/users/976/DVK-on-Ahch-To
At least part of it is probably due to the fact that Peter Pettigrew was supposed to be dead.
Voldemort and the Death Eaters couldn't talk to or about him using his real name without risking being overheard and exposed, but everyone who knew him as Wormtail was either dead, discredited, or in hiding, so it would certainly have been more convenient than coming up with a new codename, which would have taken effort he wasn't really worth.
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I figured he was called Wormtail by anyone that knew he could turn into a rat (which has a worm-like tail).
Since he spent so long as Scabbers, they likely call him Wormtail in jest at him being a rat almost as long as a human.
scifi.stackexchange.com/users/22720/Djorge
‘Wormtail’ sounded like an insult, so it’d be easy to use it as one.
Though it’s theoretically possible that the Marauders meant it affectionately, it would be very easy for the Dark Lord to start calling Pettigrew ‘Wormtail’ in an insulting way, because it inherently sounded like an insult.
Worms themselves are lowly creatures, and the name is also related to his Animagus form and connection with rats, creatures that are considered vermin.
“Wormtail has a curious affinity with rats, do you not, Wormtail?
His filthy little friends told him there was a place, deep in an Albanian forest, that they avoided, where small animals like themselves had met their deaths by a dark shadow that possessed them …” - Harry Potter and the Goblet of Fire, Chapter 33
(The Death Eaters)
However it was originally meant, it’d be very easy for a name like Wormtail to be used as an insult, as it has inherently negative connotations, so would be suited to call someone considered vermin.
The Death Eaters likely called him Wormtail as the Dark Lord did.
That the Dark Lord called Pettigrew Wormtail was likely at least part of the reason that the Death Eaters did.
When the Dark Lord refers to him to the Death Eaters, he always calls him Wormtail.
“Some of the Death Eaters he passed in silence, but he paused before others, and spoke to them.
‘Macnair … destroying dangerous beasts for the Ministry of Magic now, Wormtail tells me?
You shall have better victims than that soon, Macnair.”
- Harry Potter and the Goblet of Fire, Chapter 33
(The Death Eaters)
The Dark Lord only ever calls him Wormtail - at no point does he call him ‘Peter’ or ‘Pettigrew’, including when addressing him in front of the Death Eaters.
Since the Dark Lord always calls him Wormtail, it seems likely that the Death Eaters would then consider this the accepted way to refer to him, and would call him the same thing the Dark Lord does.
scifi.stackexchange.com/users/80602/Mal
Maybe he introduced himself as it.
I just have the hunch,
eventhough
I don't know whether DE's could use their real name 'as-is', or could have nicknames within themselves (If there is another example of DE's being called by nicknames rather than their real names, I don't know of it) Because it was a traitorous act the one he made by running to Voldemort with the potter's adderess, maybe he didn't want to be entitled also to be called by the name...
Okay scratch that!
Why would he want to be called by his nickname which will remind himself of his friends, by the very hand that killed two of them?
Masochistic and antic nature maybe?
I can't talk with much fact from here on, maybe it was a mere mockery thing from Voldemort to use his nickname, so he could bother Pettigrew with it.
I tried my best at answering...!
scifi.stackexchange.com/users/19660/Eriatta
Title says it all: I can't imagine why they would; it was an affectionate nickname given to him by his school friends, and seeing as Voldemort and other DEs view Wormtail as a coward and worthless I can't imagine why they'd use what is supposed to be, (or, what originally was) an endearing nickname for him like that.
Why do Voldemort and other Death Eaters call Peter Pettigrew his Marauder name, Wormtail?
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<harry-potter><voldemort><death-eaters>
scifi.stackexchange.com/questions/43570
scifi.stackexchange.com/users/18976/Govind Parmar
Although a nickname can be used in a friendly manner, in jest, or maliciously, one reason for using the name Wormtail may be that Voldemort and the Death Eaters couldn't talk to or about him using his real name without risking being overheard and exposed. Voldemort always used that name, and it is likely that the Death Eaters followed his example. Voldemort and the Death Eaters can’t use his real name because Pettigrew is supposed to be dead. Given that the The Dark Lord called him Wormtail, the Death Eaters did so too out of respect for the Dark Lord. 
Voldemort and the Death Eaters can’t use his real name because Pettigrew is supposed to be dead. The Dark Lord called him Wormtail and the Death Eaters did so too out of respect for the Dark Lord. 