BAM was removed from the London 2 batch of schools in May after failing to agree costs with the Education Funding Agency.
Kier will now build the five schools which are due for completion by the end of 2017.
A DfE spokesperson said: “We are working with Kier on the rebuilding of five schools in the London 2 batch.
“All are on course to be completed by the end of 2017.”
Wates has also won an £80m batch of Priority School Building Programme schools in east London.