Update as of Aug 18 2017. The below issue has now been ifixed and package application progress will be visible through the 'Apply Updates' flow in LCS.
We have discovered a bug in the 'Apply Updates' flow in Lifecycle Services, where the step details are not visible in the 'Environment Updates' section shown in the UI when a package is applied to an environment. This issue will be seen for all packages applied using the Apply Updates flow on all types of environments. The final state of the steps (completed or failed) will be updated only after the package deployment succeeds or fails which is reflected in the environment state in LCS. There is no workaround for this issue at the moment. The package deployment logs will still be available through LCS and will reflect the state of the steps on completion of the package application.
We are actively investigating this issue and will be fixing it as soon as possible. We will update the blog post once the fix is released.