The spaces you mention are symmetric spaces. The notation from Cartan's list is:

- $E\operatorname{III}$ for the quotient of $E_6$

- $E\operatorname{VI}$ for the quotient of $E_7$

- $E\operatorname{VIII}$ for the quotient of $E_8$. 

Computations for $E\operatorname{III}$ have been done in 

- H. Toda and T. Watanabe. The integral cohomology ring of $F_4/T$ and $E_6/T$. J. Math. Kyoto Univ 14 (1974), 257-286.

Computations for $E\operatorname{VI}$ have been done in 

- M. Nakagawa. The mod 2 cohomology ring of the symmetric space $E\operatorname{VI}$. J. Math. Kyoto Univ. 41 (2001), 535-556. 

I don't know of integral cohomology computations for the last one, this is more difficult due to the presence of torsion. Generally, there are a lot of computations of cohomology of Lie groups and their homogeneous spaces by the Japanese school. See e.g. Topology of Lie groups I and II by Mimura-Toda.