The exact form of the Milky Way was figured out the in early fifties of the 20-th century.
The exact form of the Milky Way was not figured out in the early 1950s. It remains a matter of debate. Astronomers agree that the Milky Way is a barred spiral galaxy. How many arms? That's still debated.
Stars in the Andromeda Galaxy were not visible to humans until recently, about 100 years ago. The Shapley–Curtis Debate occurred 101 years ago, in April 1920. Up until then, the predominant theory was that the Milky Way was the universe. This debate marked the start of modern astronomy. It wasn't until the late 1920s when Hubble's work on Cepheid variable stars convinced most (but certainly not all) astronomers that the Andromeda Galaxy was indeed well removed from the Milky Way.
It wasn't until the 1950s that radio astronomy became well-used. Infrared astronomy was even later. Radio astronomy and infrared astronomy were what enabled astronomers to see beyond the dust clouds that obscure much of the Milky Way in the visible portion of the spectrum.
It wasn't until the 1970s that distance estimates were good enough to yield a partial picture of the Milky Way. Distance remains a challenge, and without distance, there is no way to develop a picture of the galaxy. Quoting from Trigonometric parallaxes of massive star-forming regions. VI. Galactic structure, fundamental parameters, and noncircular motions, which was published in 2009,
The Milky Way is known to possess spiral structure. However, revealing the nature of this structure has proven to be elusive for decades. The Georgelin & Georgelin (1976) study of H ii regions produced what has been generally considered the "standard model" for the spiral structure of the Galaxy. However, after decades of study there is little agreement on this structure. Indeed, we do not really know the number of spiral arms (Simonson 1976; Cohen et al. 1980; Bash 1981; Vallée 1995; Drimmel 2000; Russeil 2003) or how tightly wound is their pattern. The primary reason for the difficulty is the lack of accurate distance measurements throughout the Galaxy. Photometric distances are prone to calibration problems, which become especially severe when looking through copious dust to distant objects in the plane of the Galaxy. Thus, most attempts to map the Galaxy rely on radio frequency observations and kinematic distances, which involve matching source Doppler shifts with those expected from a model of Galactic rotation. However, because of distance ambiguities in the first and fourth quadrants (where most of the spiral arms are found) and the existence of sizeable noncircular motions, kinematic distances can be highly uncertain (Burton & Bania 1974; Liszt & Burton 1981; Gómez 2006).
Astronomers did not know the exact form of the Milky Way in the early 1950s. They did not know the exact form of the Milky Way in 2009. They still do not know the exact form of the Milky Way to this day as there is a 20° gap on the far side of the galaxy that is obscured by the center of the galaxy.