But as much we know wights never show signs of decomposition and Jon used to burn bodies to save them from becoming wights.
Although we never see wights decompose in the series, there is no evidence to show that they don't decompose, given that the show doesn't follow any single wight for any significant amount of time.
However, assuming that the series follows book canon in this regard, the decomposition of a wight's remains are described very well Spoilers ahead:
A contingent of the Night's Watch was sent to the south with the still-moving hand of a defeated wight in a glass jar in order to prove that the threat is real and rally assistance in defending the Wall from the Others. However, before they could get an audience with anyone of import, the hand had rotted away to nothing:
Quote from the books:
"The same things I'd have of all of them, lad. Men, horses, swords, armor, grain, cheese, wine, wool, nails ... the Night's Watch is not proud, we take what is offered." His fingers drummed against the rough-hewn planks of the table. "If the winds have been kind, Ser Alliser should reach King's Landing by the turn of the moon, but whether this boy Joffrey will pay him any heed, I do not know. House Lannister has never been a friend to the Watch."
"Thorne has the wight's hand to show them." A grisly pale thing with black fingers, it was, that twitched and stirred in its jar as if it were still alive.
"Would that we had another hand to send to Renly." (ACok, Jon I)
"If I give them bread today, on the morrow I'll have twice as many at the gates. Who else?" "A black brother down from the Wall. The steward says he brought some rotted hand in a jar."
Tyrion smiled wanly. "I'm surprised no one ate it. I suppose I ought to see him. It's not Yoren, perchance?" (ACoK, Tyrion IV)