I love this idea. I’ll focus on New York even though I know barely nothing about it.
I think the best case scenario would be to have a social democrat come to rule New York in the 1920s and 1930s, and have them take control of housing in the city. By that I don't mean forcefully seizing land, I mean making the majority of housing council owned between 1920 and 1950. Have them build enormous structures like these in the this period to house the urban workforce:
Importantly, have the boroughs around New York reject the idea of suburbs. Have the councils massively restrict building so that no new housing can be built, and create a large green belt from Woodbridge Township through Elizabeth, Newark, Clifton, Yonkers and New Rochelle. Have Long Island do the same. Essentially this will make the housing in these areas super expensive and limit sub-urbanisation to a large degree.
Then expand affordable housing in NYC proper by building upwards. Like the enormous mountain apartments above, have buildings such as these become the norm for new builds by 1950. And so with these new builds NYC will not face a population decline or stagnation. Have the apartment buildings grow to above 20 stories, and keep on adding new floors as more people arrive in New York.
With all of that, I can see New York by 2020 having a density of 46,000 people per km2, making it the most dense city in the world, with a total population of 37 million, also making it the world's largest city.
Importantly, have apartments also be huge rather than cramped. Since the vast majority will be council housing, this will be tough. However the only way I can see this happening is by high rise technology being much more advanced to allow for a family to each get a place like that. By 2020 an average family apartment should look like this:
Firstly, this would mean that people would want to keep living in New York due to the high standard of living. Secondly it would make sure the buildings stayed nice and high to give a more 3D city feel.
By 2020, have gyms, shopping centres, indoor gardens and more be in skyscrapers, and even have vertical lifts and sky walkways to give a more futuristic feel. While not exactly what you imagined, a high rise 100 stories tall on the edge of the city would directly face small woods and semi rural housing.