Wates had been selected for the £31m job but withdrew earlier this month after failing to agree project details within the budget constraints.
The project will provide a new biosciences research centre dedicated to developing new strategies for diagnosing and treating diseases.
It is the first dedicated science building to be developed on the university's Streatham campus since 1968. It will bring together mathematicians, physicists, biochemists, cell and molecular biologists and clinical scientists in a new approach to studying sub-cellular processes, individual organisms and entire ecosystems, using techniques from mathematics and physics.
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