What would be an efficient method of global travel that can take you from, let's say, London to Hong Kong in about an hour? It would preferably be made quickly, but money is no object. I don't want something that would take 100 years to build.
Your only real option here is an Evacuated Tube transport system.
Atmospheric drag is the biggest hurdle to traveling the distance you want (9,617 km) in an hour.
So simply connect the two cities with a large diameter tube and pump all the atmosphere out of it. Then magnetically levitate and accelerate cars inside the tube. With no friction and no drag your options for accelerating the cars are only limited by your power supply and materials.
This is probably achievable with today's tech, but it would be stupendously expensive and a project of major international cooperation.
Rockets can accomplish this, but scaling it for commercial transport will be a nightmare. Factoring in fuel costs, maintenance for material fatigue from constant atmospheric re-entry, and the insurance premiums for launching passengers strapped into millions of pounds of fuel will be astronomical.
A quick calculation for a constant acceleration Evacuated Tube transit system shows that a comfortable ~3m/s^2 acceleration is required to accomplish this requirement. The cart would accelerate for half an hour, then decelerate for the last half at that rate. Energy can actually be recovered during the "slowing down" phase.
You could play with the timing by adjusting the passenger's acceleration tolerance, say spend X minutes at 1g of acceleration to get up to speed.
In London, you enter a virtual reality booth. When you open your eyes you are looking out of the eyes and hearing through the ears of a comely android with approximately your build and facial features, dressed for the season. Wearing your android body, you step out into Hong Kong.
People can know who you are as an android. Your identifiers come with when you register. You can conduct business. You can see the sights and sounds, though so far not the smells and tastes. If you do crimes as your android self (and are caught), the android will be deactivated and the London police will be waiting for you outside the booth.
To cover that distance, you need an average speed of about 2.67km/s. That's pretty brisk... even higher than the sorts of speeds anticipated of a mature vacuum maglev train technology, even if it were possible to build the required infrastructure in the given timescale (which, given the colourful politics of the countries inbetween the endpoints, may be somewhat challenging).
You could do it today, of course... modern commercial rocketry can put you into orbit in ten minutes (albeit with a certain amount of lead time to get everything ready) and from LEO you can circle the globe in 90 minutes. That'll put you half way around the world in 45 minutes, but you'll need to sustain at least a couple of gravities average deceleration on the way down which might not be much fun.
Modern suborbital spaceflight might be capable of the task... an ICBM can strike most places on earth in 30 minutes or so, though the terminal speed would indeed be terminal. Virgin's gradiosely named Galactic only reaches a mere 1.78km/s to speed and so couldn't make it, but possibly the difference could be split. Some form of hypersonic glide vehicle is probably the correct solution, with a rocket booster to get it high enough up.
The problem with all these things is that they're indistinguishable from something like Prompt Global Strike (or as previously mentioned, an ICBM). No-one wants a multi-kilometre-per-second projectile aimed at their city. There's too much scope for terrible collateral damage in the event of an accident, and with deliberate sabotage or repurposing as a missile could cause the deaths of millions. Your need for faster global travel doesn't quite tip the balance, given those downsides.
You want a hole through earth. That would allow you to go to any destination in 42 minutes, without even spending energy on the traveling itself.
Boring a hole through the earth is difficult, but if you want to go between London and Hong Kong in an hour, that is the only viable solution. It is not currently within the realm of the possible, but by some definition, it is feasible, and if money is not an object, it could be done in less than 100 years. It would, without compare, be the largest infrastructural project ever taken on by mankind.
Travel between destinations not directly opposed on the globe would take hypocycloid paths, and the tunnels should probably be evacuated, in order to eliminate drag.
In a word, Skylon.
Skylon is a reusable space plane design propelled by a SABRE air-breathing rocket engine. Test flights are expected within the next decade. Given the pace of development, British and Australian governments have started to plan a 'space bridge' between their countries. Current estimates claim a flight from London to Sydney is possible in four hours with Skylon, and London to New York in just one hour.
The main technical challenge behind the project is the need for an engine which can reduce air temperature from 1000 to -140 degrees in a fraction of a second. This allows oxygen to be captured from the atmosphere, so it does not need to be stored as liquid fuel.
Here is a technical QA with the project's staff.
Sedated, stasis transport via ICBM. Sedated to avoid the pain of high accelereation and deacceleration, in addition to the em-effects of the plasma around the capsule. Stasis, as in prepared against damage due to high g forces. May include filling the lungs with breathable fluids to prevent collaps and other protection mechanisms like metabolic cooling. https://en.wikipedia.org/wiki/High-G_training The longest part of the journey will be waking up from sedation.
There have been proposals for suborbital transports going back so long that the early ones were based on the Saturn booster (the kerosene-fueled first stage of NASA's Moon rocket). Flying at just under orbital speed, anywhere on Earth is less than an hour flight time from anywhere -- five to seven minutes of boost, at most about 45 minutes at zero gee above the atmosphere, and another five to ten minutes of the steep part of reentry and landing (this is based on vertical takeoff and landing, as with the proposed point to point service for SpaceX Starship). Passengers would remain strapped in their seats for the entire flight, and the seats would move to ensure G loads are taken in a reclined position.
With throttlable engines (like the Starship's Raptors), G load on passengers is easily limited to 3 G, perhaps even 2.5 -- this is including the Earth's gravity, not on top of it -- which is a level any passenger in good enough condition to travel ought to be able to handle in a reclined position, and far gentler than the common handling of packages by courier services.
Fuel costs wouldn't be impossible, compared to past and proposed supersonic airliners -- Raptor engines run on what amounts to natural gas and liquid oxygen, distilled from air. The premium on cost is to be anywhere, same day -- or even last night, if you're flying from the Americas to eastern Asia.
SpaceX is talking about London to Hong Kong in 34 minutes via their Starship rocket, for less than the price of first-class airfare today. But you have to add the time of the boat rides out to the launch ports. (Taking a rocket directly from Downtown London to Downtown Hong Kong is likely to run into regulatory problems.)
Nasa developed the X-43A, which can go around Mach 9.6(7,000 mph). You could build a fleet of those, but that is not cost effective. You could also equip jets with RAMJETS or SCRAMJETS to get your goal.
In my universes, there is a type of special crystal(not the point) that we use for almost everything. We achieve speeds of nearly M18 with special crystal reactor cores. The crystals are incredibly reactive, and we just compress them to get the speeds we desire.
With current technology, suborbital flight is feasible, while a hole through earth isn't.
Asking for efficiency, I would bet on some sort of Hohmann transfer (horizontal launch), plus using a cannon with external propulsion rather than the previously mentioned ICBM.
For approx. 5km/s, at 10g (100m/s2), that would require 50sec of acceleration on a cannon (rail?) length of 125km. For the 10g acceleration, you'd probably want to pack the passengers in some sort of water tank. The 125km cannon will probably be static, hence only shooting passengers from London to HongKong, and you will need some sort of reentry vehicle like the Space Shuttle, to burn away all that kinetic energy again.
Conclusion: Scuba divers in a water tank inside a Space Shuttle that's shot out of a 125km long cannon. Feasible? I don't know. The fins for the scuba divers are optional.