After running update-deployment on command prompt to deploy a custom package, if the Spotfire Analyst is started immediately, then it may not recognize the newly deployed package and hence will not prompt for an update.
Example update command:
config update-deployment --tool-password=Admin123 --area="Production" "c:\temp\CustomVisualsExample.spk"
 

TIBCO Spotfire Sever does periodic polling of the database (default value is 5 minutes) to discover this.
The same can be configured using the delay-between-refresh-minutes from configuration XML as shown below: So if we wait for this long after running the update-deployment command on TSS, the Analyst does recognize the newly deployed package and prompts accordingly.
Another option may be to restart the TIBCO Spotfire Server service.

Spotfire Analyst does not recognize newly deployed package if deployed using update-deployment command