Someone has to get serious about reusable rockets much earlier than SpaceX.
In our world, spaceflight is rare; airplane travel is common. The most significant difference between the two is that airplanes are reusable: you don't expend an airplane to travel. The fuel cost to fly something to orbit is only something like 40 times as much as the fuel cost to fly it from Los Angeles to Australia; but with disposable rockets the cost to send something to orbit is nearly four orders of magnitude greater than flying it to Australia.
The USA, obsessed with winning the space race, developed the Saturn V; this allowed the USA to send an entire moon mission in one launch. But after the excitement died down, the cost was prohibitive. Then NASA made things worse by trying to develop the Space Shuttle without anywhere near enough design/fly/test cycles. (They tried to design the first fully reusable launcher on paper without flying any prototypes, and further made it a heavy-lift vehicle! Hardly surprising that the Shuttle turned out to be not very reusable.) This mistake was partly due to severe budget cuts making it difficult for NASA to afford multiple design/fly/test cycles, IMHO a direct consequence of everyone being tired of how much money space flight cost.
If, instead of trying to win a race to the moon, the USA had tried to win a race to have a reusable spacecraft, costs would have been much reduced and we would have fuel depots in orbit and routine flights to the Moon now. We can imagine that instead of a race to land on the moon, the race was to make a space station, and they focused on making the rockets more and more reusable as time went on. They never built the Saturn V (they didn't want something like Skylab, they wanted something so big it had to go up in pieces) so they did iterate their way to something reusable. Why did it have to be so big? Maybe it needed to be an orbiting missile launch platform?
Another possible point of departure: when <PERSON> proposed the Space Defense Initiative, someone could have gotten really serious about reusable spacecraft, to launch all the space weapons and defenses. The "Delta Clipper" project, instead of being an unwanted thing NASA was forced to accept, could have been the hot new project (the project everyone wanted to work on, the one that most of the spending was for, etc.) and if lots of money and people were thrown at it could have turned into a reusable technology. 199
Just imagine that someone like <PERSON> had somehow been made head of NASA in <PERSON> first year as President, and had been given lots of resources. The DC/X did quite a lot on a shoestring budget; what could it have done if given lots of support?
Finally, I'd say you can simply imagine that a billionaire like <PERSON> had founded a company like SpaceX much earlier. That doesn't even require a particular date to make it work. (Maybe <PERSON> was even richer than in actual history, and also obsessed with reusable rockets?) The billionaire has to be willing to lose money for years until the reusable rockets start to really work, at which point the company starts charging 1/10 as much as anyone else can charge to launch things, and the company starts making huge profits and grows rapidly.
All the above could result in things like advanced space stations, or a moonbase, by 2020. But I re-read your question and saw that you are imagining millions of people on Mars by 2020. That would require reusable spacecraft much earlier, requiring substantial changes to how history actually went (maybe Rome never fell? maybe the Renaissance happened 100 years earlier? maybe the Islamic Golden Age lasted two or three centuries longer than in our history, or even never ended?)... and/or discovering some new science like antigravity or teleportation. (If you could have some kind of "stargate" like device where people could walk to Mars by walking through a gate, it becomes much easier to imagine millions of people doing it quickly.)
I just checked and according to the US FAA web site, the FAA provides service to more than 45,000 flights and 2.9 million airline passengers per day (probably a bit less now, thanks to COVID-19). Airplanes are roughly 1.2 centuries old. It's true that the flight to Mars is worse than any air travel, requiring months, so fewer people will travel by spaceship to Mars; but still, it seems safe to imagine millions of people on Mars about one century after reusable rockets become practical. So to make your timeline work with only known technology, imagine that we somehow got SpaceX-style rockets in 1920.
If reusable rockets make space travel affordable, space travel becomes routine, and colonies become possible.
<PERSON> To put colonies on Mars it would really help a lot to have some better technology than chemical rockets. 207