Yes the UK government is the largest shareholder at 49%, airline group have 46% and employees have 5%. No the government does not appoint or approve the executives.
One of the curious things about any company is that the owners make decisions, and the major owner make the most important decisions.
Amongst these you will find the appointment/approval of board members and executives.
.... and closing the airspace or eventually opening it up again
The folks down at the shop floor (or in front of that radar screen) aren't normally privy to these discussions.