Convicted child sex offenders are often considered to be the lowest of the low (i.e. the most "nonsense" people in society), and this, rather than the NONCE acronym, may be the main reason why this word became associated with paedophiles.
Calling someone "nonce" normally means one of three things:
1) UK prison slang for a paedophile.
2) A general insult, usually as an extension of 1) or 3).
3) A nonsensical person, or person behaving in a nonsensical manner, generally equivalent to stupid or idiot.
"Protect your kids from those nonces."
2) "You've been behaving like a twat all night! Piss off you annoying nonce!"
3) "He wasn't talking any sense- he was being a complete nonce!"
"Well, he's certainly nonced that up."