My child gets nervous playing in front of people.
Start small. Get your budding student to play in front of Mum or Dad, then Mum and Dad, then the family, then grandparents slowly building confidence. Also, encourage the performance of pieces that they have been playing for a long time. An easy piece played well will always sound better than a harder piece played hesitantly.
Try to encourage performance as much as possible, and the more a student does it, the easier it will become.