Every time a billionaire or government official says 'We are going to the Moon!" the actual mission gets delayed a week. So here is my sort-of educated, sort-of fanboy take on the question.
Bezos' mock up is being mocked pretty heavily by people who build space hardware. Hydrogen fuel? Really. Hydrogen sucks as a fuel in a reusable system. CO2 and water and energy makes methane and oxygen which is a much better fuel; its easy to store, needs less cooling, makes O2 which we need, gets rid of CO2, and uses a process that is very well understood. Bezos is already using metalox engines so why not use the easier to deal with chemistry? Speaking of that engine, Bezos' can barely keep his new engine from not exploding, the rocket he wants to use to lift the thing won't have the power to send it to the moon, and he is in the "guys, seriously, I promise good things are coming" phase of the typical Silicon Valley Venture Capitalist Bullshit cycle. New Glenn won't fly until 2024; I have a $50 bet on this. New Armstrong is not going to fly in the 2030's, if at all.
What Bezos has, and why I think he WILL eventually land on the moon is $100 Billion in net worth, a vision, and the desire to spend a lot of that net worth to implement the plan.
Elon Musk has a rocket that flies. He has a business with customers. He does not have a rocket big enough to launch the Orion Capsule to the moon, nor would that payload even fit ontop of a Heavy if Senator Shelby dies or retires and make the idea even possible. Dragon had a delay now, and the Boeing capsule will NEVER EVER fly on a SpaceX rocket. Musk also has no fucking clue what a timeline is. He has always, ever since Zip2, overpromised, rode his personality to the funding stage of a company, then did media song and dance to buy time until the idea works. And he's got the brains and charisma to pull that off, so good on him and it is amazing what he has done. But Starship is not going to fly in two years. Maybe four. If he does not run out of money, if he can get Dragon on track, IF he can lower costs enough to steal the rest of the launch market to build an economy of scale where he is launching 30-40 times a year and making 20-25 million in profit a launch. IF all these IF's work out, I see a Dragon on the moon in the late 2020's without a problem. R&D is expensive and you cannot just write more code to get shit moving faster... and there is still the reality that Starship is a rocket trying to build a customer base looking to create a market. the point to point idea of using his manned rockets in a suborbital fashion to move people around the earth? I have a standing bet that this will never happen. Every person closer than two miles when those engines kick in will be vibrated to death by the sound waves, and that is a bit of physics I do not see anyone, anywhere, solving. (Even ground shielding and making hills etc to cancel the noise won't work when the rocket is 300 feet in the air and accelerating.)
SLS is a mess. Mocked as the Senate Launch System, the whole rocket stack exists because of two US senators that are pushing it for the jobs in Utah and Alabama. The vehicle will run $2 billion, MINIMUM, per launch and not be reusable. If everything goes perfect, the SLS will be able to launch twice a year, max. And not be man-rated unless we throw all the safety advances on Commercial Crew out the door. SLS is eating $900 Million or so a year, and will not even launch a payload until 2021 if we are being optimistic. Then, a three year delay to the second launch which might have the Europa Clipper mission and not a moon mission as the payload.
On the whole, putting humans on the moon in five years is possible, but it will cost $4-$6 billion a year, long term commitments, and people in and out of Congress to take parentage of the project. There are people out there that hate Trump so much that they don't care about national goals, working party over country and damn the consequences. When Trump loses in 2020 my fear is that they are going to kill this program, rebrand it and start over to keep his administration's name off the accomplishment. This is also why Pence and NASA are saying five years. Do it now, do it fast and build hype before the new people can come in and fuck over your timeline.
The Lunar gateway, IMO is a waste. It's a great idea, and worth doing. Get in Lunar orbit, learn how to function out there, fine. But the Gateway is going to be some 4-5 SLS launches at $2Billion a launch, on a rocket THAT DOES NOT EXIST RIGHT NOW. The Saturn V took eight years from paper to first launch, I remind everyone, and Apollo should be thought of as a massive weapons program and not a scientific endeavor.
If I am to make a prediction, NASA will get $4 billion extra for the manned space directive over the next two years. Commercial Crew will start launching next year, in the spring, and NASA astronauts will fly on Boeing and SpaceX capsules. The ISS will go to full 12 man operations by 2020, and after that election SLS will be cancelled. the new president in 2021 will gut the lunar exploration program and want to start over, the ISS will have to fight for its life as well. First big payload to the moon will be 2026-2027 at the earliest assuming we don't gut space funding again like we have been doing for the last 30 years and it will be on a private rocket. I want this to happen. I want people on the moon to learn how to live on Mars. I want kids watching live 4K video of people on the moon. I want to see it all before I die. But the realist in me has a ton of doubts.