Appleby and Buick are the first trainer and jockey duo to win the 1000 Guineas and 2000 Guineas in the same season since Aidan O'Brien and Kieren Fallon triumphed with Virginia Waters and Foostepsinsthesands in 2005.
"I'm a very happy man, an amazing filly, what a weekend for Godolphin. To be part of this, I can't put it into words," said Buick.
"This is what dreams are made of. To me, to be the jockey on these horses is a real privilege."
Indicating the Oaks at Epsom on 6 June could be on the agenda, he added: "She's all class and may even get a little further."
Buick made all the running on Desert Flower, who is now priced around 5-2 for the Oaks.
"To have a weekend like this is very special for everybody - it's a huge team effort," added Appleby. "It was a great ride by William."