Because Alice Shrock is a goddess among Quaker Historians (especially on subjects related to Earlham through the Gurney Family), I give you this explanation:
Sir Thomas Fowell Buxton, also known as "The Liberator," was an MP who succeeded William Wilberforce in leading
the parliamentary fight for the abolition of slavery. The liberation of slaves in the British Empire was approved in 1833 and went into effect in August, 1834. Both Buxton and Wilberforce were later honored w/ life-size statutes in the nave of Westminster Abbey.
Buxton's mother was Quaker and Buxton spent much of his youth at Earlham Hall outside of Norwich. His correspondence makes it clear that the Gurneys instilled imp. Quaker values in him which shaped his career as a reformer. Although he is most known for his work against slavery, he also worked w/ EG Fry in prison reform and for the abolition of suttee in India. Buxton and Joseph John Gurney were the best of friends, and Buxton married into the Gurney family when he married Hannah, sister to EGF and Joseph John.
Buxton's income came from his work managing a brewery on or near Brick Lane; the name Truman Hanbury Buxton can still be seen on one of the buildings, I think. Also, EGF lived in the East End, and is buried at the burial grounds for Barking Meeting.
So that, I suppose, is how an East End school and street would be named Buxton, and how it would connect to Quaker Street. Your historical instincts were "right on!"