Daniell attended King Edward's School, Birmingham and proceeded to Trinity College, Cambridge (where he was the last Senior Wrangler in 1909).
For a year he lectured at the University of Liverpool and then he was appointed to the new Rice Institute in Houston, Texas.
The Rice Institute had him spend a year at the University of Göttingen studying with Max Born and David Hilbert.
In the setting of integration, he also worked on results that lead to the Daniell-Kolmogorov extension theorem in the theory of stochastic processes, independently of Andrey Kolmogorov.
[2] During World War II Daniell advised the British Ministry of Supply.