The theatre legend had said he was "furious" when Nicole phoned him a week before rehearsals for Cats began, to say she wanted to return to the X Factor judging panel. She previously appeared in Cats in the West End.
He added: "I'm furious because I really believe she's the most fantastically talented girl and I went out on a limb to get her for the London Palladium here. Never mind. There will be another girl on Broadway and Nicole will not get her Tony Award."
But Nicole, 37, has now responded via a statement issued to Radio Times, saying: "I had every intention of doing Cats on Broadway but the contract was never finalised."
She added: "I am incredibly blessed to be given so many amazing opportunities, including Cats, but unfortunately we weren't able to make it work this time around. I adore and respect Andrew, I'm so grateful for our friendship and can't wait for the opportunity to create more magic together."
Nicole did not confirm whether she is returning to X Factor. As always, bosses at the ITV talent show do not comment on judging lineup rumours until the official lineup is announced.
Former Pussycat Doll Nicole made her debut as an X Factor judge in 2012 when she served as guest judge - famously helping to form One Direction! - before landing a full judging role two years later and winning with her act James Arthur.
She also worked on the US X Factor.
There are three judging spots up for grabs on this year's X Factor as Cheryl, Nick Grimshaw and Rita Ora have all announced they won't be returning to the show alongside Simon Cowell.