From comments from @ian and @Jay:
Session timeouts are measured by checking incoming and outgoing data on the server. Unfortunately, Shiny Server doesn't surface this in a way that can be shone in a shiny app at this time.
We'd recommend looking into writing this into your application code by writing the start time to a variable to then check and display a popup when desired. You could use the shinyalert package to make a pop-up: https://rdrr.io/cran/shinyalert/
Here's the vignette and the more package info:
https://cran.r-project.org/web/packages/shinyalert/index.html