How do you know what photos look like on phones during editing on computer?

HI, Color management is not that difficult however the steps are not obvious. Try watching this short video tutorial - yes it is for Photoshop but the same concepts apply for Lightroom and it might help. The video is from Colin Smith of Photoshop Cafe. This is a great blog to subscribe to as Colin is fabulous at making complicated things more understandable. If the link does not work, just go to photoshopcafe.com and look for color tutorial.

https://photoshopcafe.com/photoshop-changes-colors-color-spaces-explained/?mc_cid=0d9f65d118&mc_eid=7757fff88d

One way I seem to always see this phenomenon is when posting to instagram.  I know you probably are not experts in the way instagram handles files, but the pictures appear this way in albums or downloaded occasionally too - this is just an easy way of showing what I'm talking about.

Here are two pictures, the first a screenshot of a lightroom file on my phone, the second a downloaded jpg of the same file, and a screenshot of the result when attempting to post the file:

https://imgur.com/a/4YpZ6Wo

The jpg experiences strange desaturation, even though the screen is clearly capable of showing all of the original colors.

Both initial pictures are identical.

Any ideas on why this is occurring?

To be truthful, the difference in displayed colors happens every day. The process of color management is rather complex, but in your specific case I would look for solution at two different places:

1. Look at your laptop's display. Is it good? Which type does it have: IPS, TFT or some hybrid stuff? Do you know the peculiarities of the technology in question? Maybe it enhances the contrast or saturates colors a lot. Maybe the color temperature is shift to blue or magenta? Usually there are tools to help with answering these questions to some extent, but a more obvious solution is to buy a colorimeter.

2. Which phones do you use to view your pictures? Here are the same questions - color temperature, contrast, saturation. Though many mid- and high-end mobile phones have really good screens in terms of viewing angles, most of them are shifted in terms of color temperature even though they cover sRGB space.

From your post it sounds as if the problem happens on many different phones, so I would tend to check the option 1.

I am not a Colour Management/Softproofing expert but much would depend on the type of screen you are viewing on an ambient lighting conditions.

For my own use (hobbyist), when I view shares  through other clients, they look fine with no desaturation. Of course, if It’s regarding critical professional use then there may be some noticeable, but then monitors should be calibrated and viewing conditions controlled.