Okay, buckle up y'all, info-dump incoming. I used to do this professionally and really like to (over)share.
First thing's first! Every movie or tv show with a decently funded post team color corrects. Every. Single. One. None of them look the way they do irl. If you've ever taken a photo or video and wondered why it doesn't look like Hollywood even though your compostion is awesome, it's probably down to two things: the lighting and the color.
Color starts on set and involves cooperation between make up, wardrobe, the art department, and the director of photography (among others). The lighting is the purview of the DP, who makes the final call on what type of light is used, its temperature, and where it's coming from. This can get complicated. Which is why it is very important to get a white balance! White balancing involves an opaque white object (or ideally a specially dedicated color card) and adjusting the color temperature on the camera so you get a true white. This is so that all colors are recorded accurately. You always want to white balance on set, or the color team will send you a giant envelope full of very fine glitter. This is mostly an exggeration, but something I did threaten my DP friend with in college.
Here's the thing: the white balance is always at least a little wrong. This is no one's fault, it's just how light and cameras work. But if it's EXTREMELY wrong.... beware of glitter.
Once you hand your footage off to your post team, lots of stuff happens, but let's focus on the colorist's job. The colorist sits in a dim room with very boring grey walls that they usually stare at every 15 minutes or so to recalibrate what neutral looks like. The colorist often forgets to blink, despite the fact that it is a necessary part of their job. Thank the colorist, for their job is necessary and under appreciated.
The first thing color does is called primary color correction, and the goal is to make sure all the shots in a single scene are as consistent as possible. Then you start on secondaries.
Secondaries is the artsy part, where you give it the Look. Last season of Game of Thrones? Famously dark. The Matrix? Famously green. Mad Max Fury Road? Famously yellow. Even in cases that are not as extreme, most movies and tv shows have a Look that is achieved by adjusting any number of factors included but not limited to: color temperature, saturation, and luminosity. These can be changed for the entire image or for only parts of it, i.e. making the shadows cooler, the highlights brighter, the midtones duller, or changing only a single object or field of the frame. Mix and match for the desired effect. This can even go so far as to adjust day-for-night or night-for-day shots, which are when you shoot an exterior scene during the wrong time of day and correct the lighting in post. They're pain, and I hate them.
In the examples above! The jogging scene probably required less extreme secondary adjustment as 'grey morning with even sky lighting' is both the lighting conditions as shot and the conditions of the final scene. In the bandstand scene, you can see what I was talking about when I say color begins on set - the strong directional light from the left side of the frame does not match at all with theĀ jogging scene. It looks like sunset. It is not. There are probably a few very strong, diffused, warm lights out of frame. That effect is something that is TECHNICALLY possible to reproduce in post without the artifical light on set, but, well...... "We'll fix it in post" is a cursed phrase for a reason. The top photo doesn't show the jogging scene's final coloring, but it does do a good job of showing the "real" colors. Take a look at Aziraphale's shirt and skin; both are significantly more saturated in the bandstand scene, the shirt almost cyan and his skin much redder, and the shadows are darkened and a bit bluer, all to give it the look of a different time of day.
Anyway. That's the basic gist of how post production color correction works. My apologies for the long post and thank you for reading.