Nadal, the 2008 and 2010 champion, was tested by world number 77 Kukushkin, a player who is coached by his wife. Williams held without fuss and, after holding three set points, then another after deuce, broke when Mladenovic strove for an ace down the middle and double-faulted.
The Serb next faces home hope Kyle Edmund, a 6-4 7-6 (7-0) 6-2 victor over American Bradley Klahn.
Dominic Inglot and Franko Skugor came through to win 6-3 4-6 7-6 (7-5) 2-6 6-2 in five sets against Fernando Gonzalez and Marcelo Demoliner.
The American won 6-4 6-4 7-6 (8/6) and will face either Dennis Novak or Raonic, whose match will be finished on Saturday.
Friday's match will however be their first meeting on grass.
Nadal will be up against the promising Alex de Minaur, who beat Pierre Hugues Herbert 6-2, 6-7 (8), 7-5, 6-3.
Her exit means eight of the top 10 seeded women have gone out before the end of Week 1.
Williams and Evgeniya Rodina are the only remaining mothers of the six who had entered this event - and they will face each other in the fourth round Monday. I always expect to come out and do the best I can do and that's all I can hope for'.
It was a comfortable victory for Britain's Murray and his Brazilian partner - winning their match 7-5 6-2 6-1 in an hour and a half.
Once the Czech survived a second-set tiebreak, she raced through the decider to make it through to the second week for the first time in seven attempts.
Since tennis turned professional in 1968, this is the worst showing at Wimbledon by the women's seeds.
Venus Williams became the latest former champion to be knocked out at Wimbledon.
Williams has dropped the opening set in both her matches so far and needed to save match points to beat Bertens in Miami this year.
It was easier to play facing a deficit, Keys said - "When you're down a set and 4-0, it's a lot easier to be, like, oh, I probably should play better now and do that" - but she admitted that throughout the final set, stepping up to the baseline to serve had been nerve-wracking every time.
Keys' departure comes hot on the heels of early exits for two-time Wimbledon victor Petra Kvitova, 2004 All England Club champion Maria Sharapova, world number two Caroline Wozniacki and US Open title holder Sloane Stephens.
Rodina, a 120th-ranked Russian qualifier, upset Keys 7-5, 5-7, 6-4. "So it will be great to play against her".