The RStudio IDE (all versions) uses the R package rsconnect to perform deployments to RStudio Connect. The rsconnect package (with packrat behind the scenes) identifies the package dependencies of your code. RStudio Connect reconstructs that environment (same R version, same package versions) on the server.
If timeDate is not found on the server, it is often because rsconnect and packrat did not identify that dependency.
Does your Shiny application include your other R file with code like the following?
source('otherfile.R')
Could you confirm that the otherfile.R is being included when you deploy your Shiny application? The easiest way to check is by looking for otherfile.R in the list of files as you go to deploy.

Could you also confirm that otherfile.R contains an explicit library statement like:
library(timeDate)
Less obvious dependencies will not be discovered.
If otherfile.R is included in the deployment and contains a simple library(timeDate) statement, we may have an issue identifying the dependencies for your content.
Could you open a support ticket for us to continue investigating? We'll want to gather details about your installation, deployment logs, and maybe the files contained in the deployment.
As a workaround, try adding an explicit library(timeDate) statement in your Shiny application code (app.R or server.R/ui.R).