The Kingdom of Kush is a continuation of the Kerma Culture, and that culture was a Cushitic one. The Nilo Saharan-speaking peoples came and mixed with the Cushitic speakers, I guess.
The people from the Kindom of Kush spoke the Meroitic language. The classification of the Meroitic language was uncertain due to the scarcity of data and difficulty in interpreting it. Some linguist claims it's Nilo-Saharan, others say Afro-Asiatic.
"According to Peter Behrens (1981) and Marianne Bechaus-Gerst (2000), linguistic evidence indicates that the Kerma peoples spoke
Afroasiatic languages of the
Cushitic branch.
[14][15] The Nilo-Saharan
Nobiin language today contains a number of key pastoralism related
loanwords that are of proto-Highland East Cushitic origin, including the terms for sheep/goatskin, hen/cock, livestock enclosure, butter and milk. This in turn suggests that the Kerma population — which, along with the
C-Group Culture, inhabited the Nile Valley immediately before the arrival of the first Nubian speakers — spoke Afroasiatic languages.
[14]"