The two SOLIDWORKS sessions are using different sets of memory to do that. This means that session 2 is not aware of the change you made in session 1. You could potentially save the part in session 1 and reload the part (File->Reload) in session 2, but I don't recommend it.
To alleviate your concern about losing work, best practice would be to save often. Use an SSD if save times are a problem.
Ctrl-S is your friend!
This is a perfect behavior in my opinion. For the session two, all the parts would open read only which have been already open in session 1. And until you not save and close the files in session 1 I don't think it would update in session 2. This is like file being used/opened in two different places and first one to open get the write access.
Try switching on the multi user option. This way you will get prompts for modified files and you should be able to rebuild to get the updated version.