He will now be in place in time for the second F1 race of 2019, the Bahrain Grand Prix on 28-30 March.
Toro Rosso have promoted deputy technical director Jody Egginton to Key's former role.
Key is the latest recruit in a major restructuring at McLaren that began last spring when it became clear how far behind the team had fallen in chassis performance.
Two of the three men who had led the design department left in 2018 - chief technical officer Tim Goss and engineering director Matt Morris - in addition to racing director Eric Boullier.
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