Ashley McKenzie, one of three Olympians from the London 2012 Games last summer on the British team in Hungary, got the nation off to a perfect start with bronze in the -60kg category.
However Britain's representatives in the women's -70kg and men's -73kg and -81kg couldn't follow suit as Conway lost to Anka Pogacnik after a first-round bye in the former.
Megan Fletcher joined Conway, who lost in the second round at London 2012, in the -70kg and won her first-round contest with Ekaterina Denisenkova of Russia by yuko.
However Dutch judoka Linda Bolder proved too much in round two while Patrick Dawson and Jan Gosiewski both lost their first fights in the -73kg to Musa Mogushkov and Sagi Muki respectively.
Tom Davis also lost his opening bout to Vitalii Dudchyk in the -81kg with the third of the British Olympians from London 2012 Chris Sherrington in action in the +100kg on the final day on Saturday.