Two points here.
Firstly, Chinooker. The procedure on a 777 is to call "Go-around, Flaps 20". Pilot monitoring (PM) retracts the flaps to 20 and waits for and then calls "Positive Rate" at which point the Pilot flying (PF) calls "Gear Up". The problem is if the PM forgets or misses the "Positive Rate" call then the gear gets left down. I suspect this is what happened to the Jet Airways you saw. It's only when they PF calls for the after takeoff checklist once the flaps are up does it get noticed (the increasing noise is also a clue!)
Secondly, GlenQuagmire. This has been discussed on here before. The usual reason for leaving the gear down after takeoff is because a wheel brake has been deactivated. This is an allowable defect. In a normal gear retraction, the brakes are automatically applied to the main gear wheels so that the wheels have stopped spinning by the time they enter the gear bay. In the case of a deactivated brake, the wheels have to be left down for 2 min after takeoff to allow the wheels to stop spinning.