By passing the EU Referendum Act 2015, Parliament authorized the Government to hold a referendum, and that referendum was consultative only. If it had intended to give the Government any power to act on the result, it would have had to say so explicitly. The Parliamentary Voting System Act in 2011 (for example) did say, in Section 8, what should be done in case of a “Yes” vote. The 2015 act said nothing about what should be done in case of a Leave or a Remain vote. It did not give the Government (or the electorate) any power whatever.
The House of Commons knew what it was doing. It had commissioned a briefing paper – No.07212, 3rd June 2015 (House of Commons Library) – to clarify matters before it voted on the EU Referendum Act. That paper says, in Section 5, Types of Referendum, “this is a type of referendum known as pre-legislative or consultative, which enables the electorate to voice an opinion” and “It does not contain any requirement for the UK Government to implement the results”.
The Government knew that perfectly well, too. It knew that it was not telling the truth when it presented the referendum as a “decision”.