I know for my Transcontinental Latitude passes I need to first be able to book space in B class to get on the plane and then as long as there is J > 0 I can upgrade. Since at 48hrs there is a better chance of J > 0 than R > 0 (for your upgrade certs), the upgrades tend to be a bit easier to get (granted I have no experience on the LHR passes).

Those are all my tricks. I'm sure others have more.