The men, playing at the Diners Club in Slovenia, beat Italy 3-2; while the women, playing at Sierra Golf Club in Poland, defeated Sweden 3.5-1.5.
England men’s captain Roy Smethurst commented: “The team has done very well. Last year we were fifth, this year we’re third – next year hopefully we’ll be first.”
Yorkshire’s East put the men on the route to the bronze medal when he polished off his singles 7&6, playing the 12 holes in five-under par.
“He played quite brilliantly,” said Smethurst.
David Niven (Newbury & Crookham) then powered to his first singles win of the week, defeating his opponent 6&4, while Stoneham's Alan Mew sealed the win with a superb chip on the 15th to sign off 4&3.
The men qualified in third place for the match play stages of the championship and defeated Spain in the quarter-finals. But their hope of winning the gold medal was dashed when they lost to the eventual champions, Scotland, in the semi-final.
The women were top seeds in their championship and defeated the Netherlands in the quarter-finals before they, too, were beaten in the semi-finals, losing to Belgium.
The bronze medal play-off pitted them against Sweden and the match started with a tight foursomes, but Leicestershire’s Helen Lowe and Richmond's Karen Jobling played par golf to win 3&2 and set England on the path to the medal.
In the singles, English senior amateur champion Cath Rawthore went two down early in the match, but played steady golf to fight back to level terms and go on to win 3&2.
Behind her, English senior stroke play champion Jackie Foster lost an early lead of two holes and was all square after 13. But she won 14 and 15 to be two up on the 16th, where she just missed her birdie two, and she won her match at the 17th, securing the bronze medal.
British senior champion Julie Brown was two up playing the 17th but her match was declared a half as England had sufficient wins to claim the medal.
Team captain Wrightson said: “The standard of golf has increased significantly this year and the all the matches in Flight One have been a challenge.
“The England players dug deep to pull out their best golf to win the bronze medal after the disappointment of losing in the semi-final, having been leading qualifiers.”