To date, there is no evidence that the Christian hymn, "Amazing Grace," traces its melodic structure to Cherokee Nation flute songs. The lyrics of "Amazing Grace" were composed by Anglican clergyman, John Newton in 1779. However, it was not until 1847 that the "New Britain" tune that is associated with "Amazing Grace" became popularized by William Walker, a composer.
The Trail of Tears occurred between 1831-1850. So there is a slight possibility that William Walker was given the tune from someone who witnessed or took part in the Trail of Tears forced migration. In fact, William Walker was from Spartanburg, South Carolina, which before the Trail of Tears was part of the Cherokee Nation territory. The removal of the Cherokee in Spartanburg, South Carolina began in 1835. That same year, Walker published a tune book titled, The Southern Harmony. The revised 1847 version of The Southern Harmony was where the modern "Amazing Grace" tune first appeared.