Since you already have all the data in a Google Sheet, I think the easiest way to get this working automatically is using Google Apps Script. I will leave an example of how it would work.

Try this code:

function myFunction() {
  var ss = SpreadsheetApp.getActive().getSheetByName("Sheet20"); // Change it to the name
                                                                 // of the sheet you. are using
  var range = ss.getDataRange();
  var values = range.getValues();
  for(var i=1; i<values.length; i++)
  {
    values[i][1] = lastModifiedTime(values[i][0]); // 0 is the column number in my testing sheet
                                                   // where the IDs are.
                                                   // 1 is the column number where the time will be.
  }
  range.setValues(values);
}

function lastModifiedTime(id)
{
  var file = DriveApp.getFileById(id);
  return file.getLastUpdated();
}

Input data:

Replace the Sheet ID fields with valid IDs.

Result:

Remember to change the format of the column where you want you have the last modified time to Number > Date time.

Now, in order for you to automate this script you can use a time-driven trigger from Google Apps Script so that the script runs every 5, or 10 minutes for example and so that you can always get the latest time in the results.

Trigger:

References:

• Method: getLastUpdated()
• Time-driven triggers