Alternatives
An answer may be derived from observing the progress of commercial aviation in Japan. After the Narita debacle, the Japanese have only built large airports on reclaimed land. There are many problems with this approach (new Doha airport an example), but not as many and displacing people and confiscating land. The new runway Haneda airport is built on reclaimed (more accurately; generated) land in Tokyo Bay. No such option exists for LHR.
England is a relatively population high density country, especially in the vacinity of London. A more practical approach to expanding commercial aviation capacity in the area would likely be to seriously consider an airport on new land. Such a decision might aggravate those living in coastal communities near to the new facility, but it would offer the benefit of unimpeded approaches and departures and 24 hour operations. There are many good (and not so good) engineering firms with vast experience building such airports. It may be the most cost efficient and hopefully, politically acceptable solution. This approach would also provide vast employment for quite a few years.
As many governments have found out, airport projects tend to generate the NIMBY reaction (Not In My Back Yard). It may not generate quite so much sympathy if a few complain of it impeding their ocean view. Will likely be MORE difficult satisfying the environmentalists!