They should be half way through building 'Boris Island' by now! 
They wouldn't have even begun the 20year long feasibility study yet, on year 18 of the study, after having spent £92bn on it so far, they will begin to consider flight and landing routes, transport systems to the airport and where the workforce will live and relocate to, so in year 19 of the study it is cancelled due to:
1. Dutch refusing to allow interference with Schiphol flight routes and landing patterns which was known at the start but they assumed the Dutch would cooperate!
2. The feasibility studies for the building of the transport systems will take at least 20 years and at least another 20 to get the planning and environmental studies done another 20 years to purchase the land and get detailed designs done and then allow 40 years to build them. If you think I'm joking just study HS1 and even more HS2. Estimated completion date is 2199, where Boris Island was targeted at 2120 to open*.
3. Local objections to the building of 200,000 houses in the vicinity to house the new airport's workers, so they will have to do a 200-mile daily commute from where they currently live until the house are built beginning with an estimated 2180 start.
4. Unions indicate they want at least £100k disruption and relocation cost per member of staff moving and the guaranteeing of all jobs!
5. The country can't afford the estimated £4tn cost and supersonic trains like the Hyperloop etc. means by the time it opens flying will be more expensive in many cases. Elon Musk wants to build 2 in the UK from London to Manchester and London the Edinburgh. London to Edinburgh journey time is estimated to take 20mins.
6. Studies of the SS Richard Montgomery show the cargo is highly corroded and the ammunition is highly unstable and if it blows the resulting tsunami would flood Boris Island and central London.
*Since Heathrow opened in 1946 (71 years ago) there has been no direct rail link from the West. A loop off the GWR line was suggested about 10 years ago. This has just progressed from a feasibility study to detailed planning. It is expected to be completed in the mid to late 2020's. This is only a few miles long and shows the glacial speed that Government infrastructure projects proceed at. Currently you have to travel into London and then back to Heathrow or use a bus from the West.