In October 2020 the Court of Appeal ruled in favour of the campaigners claim that development of the two mile 3rd runway breached the government’s policy of removing almost all carbon emissions from the economy by 2050 based on a global temperature rise of 2C.
In December 2020 this judgement was overturned by the Supreme Court which regarded the policy as being legitimately based on the previous less stringent temperature target of 1.5C with implications for emission levels.  However, the 3rd runway still requires a development consent order,that is to say Planning Approval for Large Projects.
Another option that may be favoured by the government is to expand airports elsewhere without extra runways.
Heathrow Appeal