Departure runway 21 calls for a right turn to the sea at 400', thence SID to Chekka (CAK) which is 030 track. This path crosses final rwy 16 at low altitude, so after 5000' ATC rightly tried to vector towards the West to avoid approaching traffic. Then they did an almost complete left two-seventy back to South of the airport, climbing to 9000 feet (4000 feet is 1 1/2 minutes). Continue on this path would have flown them directly into the mountains. EGPWS warning adding to confusion?
My speculative bet - autopilot disengagement went unnoticed when switching to heading mode - tried manual correction while spatially disorientated = Kenya + Flash