Developing black and white 35mm film in the 1920s would have been similar to how it's done today. Here is what I found, along with links...
It seems you've already done some research to determine that 35mm cameras, such as Leica A, were available in the 1920s. You can read a bit about 35mm film on Wikipedia (135 film; 35mm movie film). Note that preloaded cassettes were not available until 1934:
- In the earliest days, the photographer had to load the film into reusable cassettes and, at least for some cameras, cut the film leader. In 1934, Kodak introduced a 135 daylight-loading single-use cassette. This cassette was engineered so that it could be used in both Leica and Zeiss Ikon Contax cameras along with the camera for which it was invented, namely the Kodak Retina camera.
Wikipedia also has a Timeline of photography technology and List of discontinued photographic films that might be helpful.
- 1922 – Kodak makes 35 mm panchromatic motion picture film available as a regular stock.
This film was developed using the gelatin silver process that's still used today. The process was developed in the 1870s, and Kodak has been selling films, papers, and chemicals since the 1900s. However, I don't know the form or portability of the chemicals.
Color photography is likely outside of the scope of your story, but you might be interested in reading about it anyway. A Quick History of Color Photography (for Photographers). Kodachrome was introduced in 1935, and the standard C-41 process still used today was introduced in 1972.
Development uses a fair amount of water. According to How did living standards change in the 1920s?, "By the 1920s most small cities had paved streets, municipal electricity and water systems, telephone systems, streetlights, and sewage systems..." So you wouldn't have to worry about bringing in and disposing of buckets of water.
Early Photography: Darkroom Equipment has some interesting information. Although I don't see one for 35mm film, daylight tanks were available for other film formats in the 1920s, so there might have been one for 35mm film. It might also be possible to have developed 35mm film in a 127 film tank.
Initial loading of the tank would have been done in a darkroom or changing bag. I don't know when changing bags were invented. However, the creation of a makeshift darkroom could be done similarly as would be done today. Lock oneself in a closet and tape off any light leaks. While Gaffer tape does not appear to have been invented until 1959, "duck" tape (made with duck cloth) was available in 1902.