Workers are expected be given 16 weeks to have the jab - or risk losing their job, the BBC understands.
The Met's top officer Dame Cressida Dick has apologised to the Morgan family in the wake of the report.
A clip shared on social media shows Newsnight's Nick Watt being harassed by anti-lockdown protesters.
A number of major companies say they are pausing adverts as they review the news channel's content.
The whole country had been due to move to the lowest level zero restrictions on 28 June.