Williams, 18, led with just 10 seconds remaining, but a ferocious late attack by opponent Mengyu Zhang saw the Chinese fighter win 31-22.
Longobardi-Radford, who has previously competed for Spain, lost 24-10 to Yu-Jen Huang.
"I just didn't get my tactics right," he told BBC Sport.
Victory in the quarter-finals would have guaranteed the British fighters a medal.
Williams, a double world junior champion and senior European gold medallist, opened her day with a dominant 29-7 defeat of Ivory Coast fighter Marie Federique Ekpitini.
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The seventh seed followed that up with a tense sudden-death, golden-point success over Athanasia Mitsopoulou of Greece, before a concentration lapse saw her eliminated in the next round.
Longobardi-Radford, who had won all five lower-ranking events he entered since joining the Great Britain programme in 2016, opened with a 18-11 preliminary round defeat of Turkey's Ferhat Can Kavurat.
Emphatic defeats of Moroccan fighter Faical Saidi (19-4) and Arven Al Cantara of the Philippines (22-8) followed before he was outclassed by Huang of Chinese Taipei.
"For sure I'm happy with what I've achieved recently, but today wasn't the performance I was looking for and that's a disappointment," he said.
Defending +73kg champion Bianca Walkden, who two years ago became only Britain's second World Taekwondo gold medallist, begins her campaign for a further title on Tuesday.
Fellow Team GB Rio Olympic medallists Lutalo Muhammad and Jade Jones begin their quests for their first world titles on Wednesday and Thursday respectively.
Britain's Lauren Williams and Peter Longobardi-Radford missed out on medals after losing in the World Championships quarter-finals in South Korea.