OK, I called CO and asked about the star upgrade awards.

The speed with which they can tell you if an upgrade is available or not will depend on the partner airline. They said, some partners such as UA, LH will come back in a couple of minutes with the status of the award.

Also another important point to note is:
When you buy the Y/B fare, the flight number should be the actual airline flight number.

Therefore, if you fly LH, then you need an LH flight number...not a CO-codeshare number of the same LH flight.

Due to the above, it becomes pretty hard to find the correct fares on co.com, which IMO is poor in showing partner flights in general.