Wales beat France 19-12 in the quarter-finals but went down 21-19 to South Africa in the semi-finals.
South Africa went on to beat Fiji 24-14 in the Cup final while New Zealand, beaten 42-19 by Fij in the semi-final, had nevertheless secured enough series points to clinch the World Series title.
South Africa's victory in London means the race for second place in the Series behind the Kiwis will be a four-way battle at next weekend's Emirates Airline Edinburgh Sevens involving England (121), South Africa (116), Samoa (112) and Fiji (110).
Meanwhile, Scotland claimed four Series points by beating Kenya 21-19 in the Bowl final and England ended a disappointing weekend by beating Portugal 22-7 in the Shield final.
Gareth Davies, Jevon Groves and Alex Cuthbert were the Wales tryscorers in the win over France.
South Africa took control of the semi with tries from Johannes Prinsloo (2) and Bernado Botha. Groves and Cuthbert and Kristian Phillips scored Wales' tries but Gareth Davies could not convert the latter one to take the tie into extra time.
Scotland beat Spain 19-17 to reach the Bowl final with tries from Peter Horne (2) and Andrew Turnbull.
In the final Turnbull's late try levelled the scores and Horne's conversion clinched victory.
England, having suffered a shock 19-17 defeat against Kenya at the start of the day when Lavin Asego's last-gasp try sank them, beat Canada 22-14 with tries from Nick Royle, James Rodwell, Dan Norton and John Brake to set up their Shield final meeting with Portugal.
England took control of the final with first-half tries from Mat Turner, Norton and Brake in the first half. Norton claimed his second try after the break.
England head coach Ben Ryan said: "With a good performance in Edinburgh it may still be our best overall performance in the 12 years of the series."
The Plate final was won by Samoa, who beat defending London champions Australia 22-12.
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