In Harry Potter and the Deathly Hallows, Ron loses his wand during the capture and subsequent imprisonment in Malfoy Manor. When they take their opporunity to escape, they also take with them Wormtail's wand:

‘And we’ll have that,’ whispered Ron, tugging Wormtail’s wand from his other hand.

^{(HPDH - Chapter 23 - Malfoy Manor)}

When speaking to Ollivander later, Ron asks whether he should use Wormtail's wand:

‘So I should use this one?’ said Ron, pulling Wormtail’s wand out of his pocket and handing it to Ollivander. ‘Chestnut and dragon heartstring. Nine and a quarter inches. Brittle. I was forced to make this, shortly after my kidnap, for Peter Pettigrew. Yes, if you won it, it is more likely to do your bidding, and do it well, than another wand.

^{(HPDH - Chapter 24 - The Wand Maker)}

We take it that Ron uses Wormtail's wand throughout the Battle of Hogwarts. Once the battle is over and the wizarding world returns to relative normal, is there any reference from beyond the books that mentions whether Ron continues to use this wand?

Did Ron continue to use Wormtail's wand?

Additional thoughts:

Other characters have expressed concern over using particularly notable wands:

Hermione looked frightened that the wand might sting or bite her as she picked it up. ‘I hate this thing,’ she said in a low voice. ‘I really hate it. It feels all wrong, it doesn’t work properly for me... it’s like a bit of
her.’
[...]
‘It’ll probably help you get in character, though,’ said Ron. ‘Think what that wand’s done!’ ‘But that’s my point!’ said Hermione. ‘This is the wand that tortured Neville’s mum and dad, and who knows how many other people? This is the wand that killed Sirius!’

Presumably, if Hermione ended up with Bellatrix's wand she may not have kept using it even if it had yielded to her.

It’s likely Ron got a new wand afterward.

Though Ron used Wormtail’s wand while there was a war going on, it’s likely he’d have wanted his own when things settled down. In general, wizards tend to prefer a wand that’s chosen them to a secondhand one.

“Most witches and wizards prefer a wand that has “chosen” them to any kind of secondhand wand, precisely because the latter is likely to have learned habits from its previous owner that might not be compatible with the new user’s style of magic.”
_{^{- The Tales of Beedle the Bard}}

Though wizards can use wands that haven’t chosen them, one that did is always going to produce the best results.

“The wand chooses the wizard,’ said Ollivander. ‘That much has always been clear to those of us who have studied wandlore.’

‘A person can still use a wand that hasn’t chosen them, though?’ asked Harry.

‘Oh yes, if you are any wizard at all you will be able to channel your magic through almost any instrument. The best results, however, must always come where there is the strongest affinity between wizard and wand. These connections are complex. An initial attraction, and then a mutual quest for experience, the wand learning from the wizard, the wizard from the wand.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 24 (The Wandmaker)}}

While Wormtail’s wand worked better than a random wand for him, that doesn’t mean it’ll work better than a new wand that chooses him. Though it’s true that Ollivander tells Ron to use Wormtail’s wand, that’s because Ron’s own wand was taken by the Snatchers, and Ron asks him which of the wands they’d stolen he should use. Having a wand choose him wasn’t an option. When he says it’s more likely to work than another wand, he meant a random wand that didn’t choose him or be won by him.

“I took this wand from Draco Malfoy by force,’ said Harry. ‘Can I use it safely?’

‘I think so. Subtle laws govern wand ownership, but the conquered wand will usually bend its will to its new master.’

‘So I should use this one?’ said Ron, pulling Wormtail’s wand out of his pocket and handing it to Ollivander.

‘Chestnut and dragon heartstring. Nine and a quarter inches. Brittle. I was forced to make this, shortly after my kidnap, for Peter Pettigrew. Yes, if you won it, it is more likely to do your bidding, and do it well, than another wand.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 24 (The Wandmaker)}}

Further proof of this is in Harry’s experience. Draco’s wand which he won from him worked about as well for him as Hermione’s did, but not as well as his own wand did, since Harry very much wanted his own wand back.

“Harry looked down at the hawthorn wand that had once belonged to Draco Malfoy. He had been surprised, but pleased, to discover that it worked for him at least as well as Hermione’s had done.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 26 (Gringotts)}}

So, it’s likely, in most cases, wizards would want a wand to choose them if they need to replace their own wand. In Ron’s specific case, he’s even more likely to want a new wand, since he’d been using the wand he took from Wormtail, who was a coward that pretended to be his pet. Ron was revolted by him. It’s likely he wouldn’t want to use Wormtail’s wand as his own wand after the war was over.

“Ron … haven’t I been a good friend … a good pet? You won’t let them kill me, Ron, will you … you’re on my side, aren’t you?’

But Ron was staring at Pettigrew with the utmost revulsion.

‘I let you sleep in my bed!’ he said.”
_{^{- Harry Potter and the Prisoner of Azkaban, Chapter 19 (The Servant of Lord Voldemort)}}

As mentioned in the question, Hermione didn’t want to be around or use Bellatrix’s wand - it’s likely Ron would have some degree of revulsion to Wormtail’s wand. It also doesn’t seem like Ron would be too poor to afford a new wand so would be forced to use it out of necessity. J.K. Rowling mentions in an interview that Ron, along with others like Harry and Neville went into the Auror Department right after the war. He’d likely be paid well enough for that to buy himself a wand.

JKR: Of course she'd go back. She has to get her N.E.W.T.s. Ron was really done with schooling. It would be kind of tempting to go back just to mess around for a year and have a break, but he goes into the Auror department. He's needed. Anyone. Anyone who was in that battle on the right side, Kingsley would want them to help clean up the-- I mean, anyone who's old enough to do it, who's over-age. But Kingsley would've wanted Ron, Neville, Harry and they would've all gone, and they would've all done the job. And I think that that would've been a good thing for them, too. Because to go through that battle and then be religated to the sidelines, I think they would've felt a need to keep going and finish the job. So that would've been rounding up, really, the corrupt people who were doing a Lucius Malfoy and trying to pretend that they weren't really involved.
_{^{- Interview with J.K. Rowling (17 December 2007)}}

Plus, he helped free Ollivander from being held in Malfoy Manor, so it’s possible he could get a free wand. Ollivander sent Luna a free wand for being good company - if Ron asked him for a wand, he’d likely get it.

“Mr Ollivander had sent Luna a new wand that morning. She was out on the back lawn at that moment, testing its capabilities in the late afternoon sun.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 26 (Gringotts)}}

For these reasons, though it’s never explicitly stated that Ron got a new wand, it seems very likely he would have.

He most likely continued to use Wormtail's wand.

Since Ron had lost possession of his previous wand when it was taken from him by the Snatchers, it no longer "belonged" to him.

Harry was still half blind, barely able to see anything through his puffed-up eyes. When at last the man tying them up had walked away, Harry whispered to the other prisoners.

"Anyone still got a wand?"

"No," said Ron and Hermione from either side of him.
_{Harry Potter and the Deathly Hallows, chapter 23: "Malfoy Manor"}

And, in the quote you provide, Ollivander tells him that's more likely to work for him than any other wand:

"So I should use this one?" said Ron, pulling Wormtail’s wand out of his pocket and handing it to Ollivander. "Chestnut and dragon heartstring. Nine and a quarter inches. Brittle. I was forced to make this, shortly after my kidnap, for Peter Pettigrew. Yes, if you won it, it is more likely to do your bidding, and do it well, than another wand."
_{Harry Potter and the Deathly Hallows, chapter 24: "The Wand Maker" (emphasis added)}

I also vote for Ron continuing to use Wormtail's wand, but for more practical reasons: We know the Weasleys aren't exactly well-off, and wands are expensive - so I can see Ron thinking to himself, "Hey - Free wand!".