Should you have an appointment at Heathrow to see the future of flying, let me offer a tip: don't fly there. My plane touched down on the northern runway on time, but there was no spare gate at Terminal 5. (This is the £3.4bn structure that, you may recall, was supposed to ease the squeeze at Heathrow.) So the Airbus pilot took a meandering route to a remote stand well south of the building. Then a no-frills bus took us on another tour of the apron, back to the north of Terminal 5.
British Airways uses all three odd-numbered terminals at the airport, which makes for some interesting transfers. I didn't check if any of the passengers booked on other BA flights had tight connections, because I was sprinting to the platform for the rail link to Heathrow's centre. I missed the train by seconds and had to wait another 15 minutes. So I was late meeting the team that is resurrecting LHR. But at least this was one of those rare occasions when you might expect the people you are meeting to apologise for your tardiness; it's their airport, after all.
The delay summed up Heathrow 2013: even when the airport is free of disruption from snow, fog or ash, the traveller can feel trapped in a treacle of inadequacy. Like many of its passengers, Heathrow struggles with some ungainly baggage.
Airports are like hospitals and prisons. Any sane person wants to get out as soon as possible – preferably on a jet to somewhere lovely, otherwise on a train home. That is particularly true in the curious assortment of buildings, taxiways and tunnels that, in the case of Heathrow, provide a good definition of the word "motley". For seven decades, London has shambled its way towards the future of flying, using an infrastructure held together by Sellotape, string and some astute cramming of a quart into a barely adequate pint pot.
Mind you, we haven't done badly. By an aeronautical mile, London is the capital of world aviation. Add Gatwick, Stansted, Luton and London City passengers to Heathrow's 70 million a year and you reach 130 million. That's way ahead of New York, Paris, Tokyo and every other world city. And all this on five-and-a-bit runways (London City is the puny one), which Amsterdam Schiphol alone can almost match.
Even without another runway in South-east England, there's room for many more people. Luton is half-empty. So is Stansted – acquired this week by Manchester Airports Group, which must boost traffic dramatically to justify its £1.5bn investment. Gatwick, the world's busiest single-runway airport (only San Diego comes close) could handle one-third more passengers.
Inconveniently, though, for London's "challenger" airports, passengers value Heathrow above all. "Airlines make more money flying to Heathrow," says Colin Matthews, the airport's chief executive. The most precious commodity in aviation is an intangible: permission to take off and land on one of Heathrow's two runways.
You know, of course, the unique burdens under which Heathrow labours. The basic plan for Europe's busiest airport was outlined 70 years ago as the Second World War was ending. Today, Heathrow is hemmed in by housing, a couple of trunk roads and the M25, compressed into a site absurdly close to central London. Congestion plus the prevailing westerly wind oblige airliners to circle over the Home Counties, pirouette around the Shard and annoy a couple of million people as they descend over Western Europe's biggest city.
When you touch down, your problems may only just be starting. Terminals 1 and 3 looked good in the Sixties, but now feel hopelessly outdated. Terminal 4 was an ill-judged 1980s stop-gap shoehorned into an awkward corner. All these older terminals were built with culs-de-sac in which aircraft can be expensively trapped while passengers fume and accountants frown.
Terminal 5 was supposed to put everything right, but it suffered from the longest planning inquiry in history and went wrong. BA's home was designed for 20th-century air travel. By the time T5 opened in 2008, there were too many check-in desks and not enough gates.
At least T5 is the right shape. Together with its two slim satellites, it sits neatly between the runways, demonstrating that the future of airports is … the toast rack. "It's really efficient," says Duncan Pickard, project director of Terminal 2. With terminals parallel to each other and perpendicular to the runways, aircraft are never boxed in.
Mr Pickard has the unenviable task of creating a vast and sensibly aligned cathedral of aviation plumb in the middle of Europe's busiest airport. Right now, it is Britain's biggest building site: "We have 2,000 people a day working here".
The original T2 was one of the earliest grand projects officially opened by that other great treasure from the early 1950s, Queen Elizabeth II. Her Majesty has endured much better than the old airport terminal, which was flattened (along with the neighbouring Queen's Building) almost as soon as T5 was running properly.
Mr Pickard has a scale model of the project. T2A, which is the main element, looks squat and square, as though it were a product of the Lego school of architecture. Its daughter, T2B, is a much thinner slice of toast a few hundred metres east, connected by subterranean tunnels.
To find if the reality is more inspiring than the papier-mâché, we donned hard hats and some dainty gloves and headed into the future.
And what a prospect it is. Whether you arrive by car or train, your journey starts at the top. Unlike every other airport, the check-in area is refreshingly uncluttered. That's because the designers know that most 21st-century passengers don't check in, at least in the traditional sense. Instead, ranks of automatic kiosks await those passengers who didn't get around to checking in online. There are some "classic" desks, but most people will simply want to drop a bag.
The typical T2 passenger will have plenty of baggage. The main tenants are the members of the Star Alliance, which includes United (currently split between T3 and T4), Singapore Airlines, South African Airways and Air Canada. Long-haul passengers tend to carry more than short-haul travellers – though there will be plenty of those, with Lufthansa and Aer Lingus also in residence.
Your luggage will go off on its own little adventure, thanks to an intriguing Heathrow fudge. Terminal 1 has more than enough capacity to screen all the passenger luggage for T2, so every piece of baggage gets sent along an elevated passageway to an entirely different part of the airport. Here it is examined and loaded into containers that are then driven back around to T2.
While they aren't checking in or wondering where their luggage is, passengers may contemplate the covered court that separates the car park and terminal – a necessity dictated by fears of a terrorist attack with a vehicle bomb. It will feature a 70m, 70-ton sculpture called Slipstream.
The next stage of the journey is also determined by terrorism: the biggest security search area you ever did see, wide and deep enough for the longest queues. Not, Mr Pickard hopes, that there will be significant delays: "Economy passengers will have 17 lanes; there are four 'fast tracks' for business passengers and three lanes for staff."
Next, natural light draws you home, or at least to the area where you are expected to spend most of your "dwell time". And while it will look a little different during the 3,200 passengers-per-hour peak, this is genuinely a space that lifts the spirits. It's a more human environment than the average airport, partly because it is bathed in daylight. The curving roof has slices of glass that lure in natural light from the north – avoiding direct sun and overheating.
"You're not in a box," says Mr Pickard. "This is all about a great passenger experience. The airlines are thrilled that they can bring people here, and staff will have a happier environment."
Intuitive wayfinding means passengers should waft easily towards the departure gate. When the cabin door closes, the plane pushes back unimpeded from the toast rack to start the real business of travel. Elegant it may be, but T2 is just a means to an end. For arrivals, the touchdown-to-Tube escape looks easy (passport queues permitting). Croissant-shaped baggage carousels enable the passenger payload of an Airbus A380 to crowd around to watch the hold's contents arrive. Once reunited with baggage, passengers accustomed to the original T2 get a momentary flashback; the old tunnel to the Piccadilly Line has been retained.
A treat awaits domestic passengers. Arrive from Manchester, Edinburgh or Aberdeen with an onward flight from T2, and you can access the departure lounge without enduring another search; arriving from another UK airport means you are deemed "clean". The trouble is: the facility was planned for BMI. But BA bought the airline last year and is moving all mainland domestics to T5.
Virgin Atlantic is starting UK flights on Easter Sunday. Heathrow has offered space for them in T2 next year, but Virgin has every intention to co-locate the links in T3 alongside its international operation.
After the T5 fiasco, in which the new terminal seized up within a couple of hours of opening, T2 will roll out slowly, starting about a year from now.
And the £2.5bn cost? That's down to you and me, with passengers paying the highest airport fees in Britain – more than double the rate Gatwick charges, and set to rise steeply over the next five years.
"This is about keeping Heathrow competitive for passengers and airlines over the next five years," says Heathrow boss Colin Matthews. "Making sure passengers choose to use Heathrow instead of Paris or Amsterdam or Frankfurt, or indeed Dubai or Istanbul."
Could T2 be the new aeronautical Jerusalem in England? Let's hope so, or the UK could be toast.
Heathrow: the rivals
Heathrow handles 70 million passengers a year. The airport's boss cites these as his competitors. Their passenger numbers are expressed as a percentage of Heathrow's score.
Schiphol may be the home of the world's oldest airline, KLM, but it is a testament to deft, organic planning. In a single terminal it handles 67 per cent of Heathrow's throughput, and squeezes in Europe's best airport shopping, plus an annexe of the Rijksmuseum.
Charles de Gaulle's Terminal 1 looks like a sci-fi throwback, but the main Air France operation at T2 is one of Europe's leading hubs. And while the main Paris airport scores only 81 per cent of Heathrow, there's also a big shed, T3, for no-frills flights. They wouldn't do that at LHR.
Lufthansa's main base achieves 77 per cent of Heathrow's capacity, but within a much more coherent huddle of terminal buildings – and with a proper high-speed rail station as an integral part of the airport. That beats the Piccadilly Line.
"Connecting the world," boasts the airport slogan. Emirates' "old" airport is chasing Heathrow, already achieving 75 per cent, with a new dedicated Airbus A380 terminal just opened. Oh, and in a spare patch of desert there's a six-runway airport getting ready for take-off.
Turkish Airlines lost the plot for a while and let its Gulf rivals pinch much of the north and west/south and east traffic. Now it's fighting back, with 60 per cent of Heathrow's numbers at the existing Ataturk facility and plans for a huge new airport to match the best in the Arab world.