Doctor Who needs its actors on set in Cardiff, Wales for eight months, and Cumming just wasn’t willing to relocate for such a long period of time.
Back in the early days of the revival, Alan Cumming was approached by then-showrunner Russell T. Davis about playing the Doctor.
So, the Russel T. Davis-era Doctors were Christopher Eccleston and David Tennant, with Alan Cumming nowhere to be seen.
Cumming was amenable to the idea up until he learned that doing so would require him to move back to Britain from New York, where he’d been based for many years.