Unclear, but he won’t have been convicted as he was Imperiused.
Though it’s never explicitly stated, it seems likely that Thicknesse would have survived the battle, as his death is never mentioned, and as the Order knew he was under the Imperius Curse, it was unlikely they’d particularly want to kill him rather than just stop him from fighting. There’s no real information about what he did after the battle. The one thing that can be reasoned from what is known is that, presuming he did indeed survive, he wouldn’t have been convicted over his actions as Minister. The Order knew he’d been Imperiused, which would mean Kingsley knew as well.
“The official version of Scrimgeour’s murder is that he resigned; he has been replaced by Pius Thicknesse, who is under the Imperius Curse.”
- Harry Potter and the Deathly Hallows, Chapter 11 (The Bribe)
Avery was able to get himself acquitted from his crimes by claiming he was under the Imperius Curse, showing that being Imperiused is considered a reason for acquittal.
“Avery – from what I’ve heard he wormed his way out of trouble by saying he’d been acting under the Imperius Curse – he’s still at large.”
- Harry Potter and the Goblet of Fire, Chapter 27 (Padfoot Returns)
Once someone is known to be under the Imperius Curse, they’re not convicted. After the first war with the Dark Lord, the Ministry of Magic had to figure out who had been Imperiused and who hadn’t been, so they could charge the people who willingly acted but not those forced into it.
“Years back, there were a lot of witches and wizards being controlled by the Imperius Curse,’ said Moody, and Harry knew he was talking about the days in which Voldemort had been all-powerful. ‘Some job for the Ministry, trying to sort out who was being forced to act, and who was acting of their own free will.”
- Harry Potter and the Goblet of Fire, Chapter 14 (The Unforgivable Curses)
Thicknesse was known by the Order to have been Imperiused, and Kingsley, one of the Order’s members, became Minister and oversaw the efforts to arrest the remaining of the Dark Lord’s supporters. Once Thicknesse survived, he wouldn’t have been convicted - though it’s unclear what did happen to him and it’s never mentioned what (if any) life he led after being Imperiused.