AA/UA: Companions can be upgraded at your same upgrade window.

CO: Gold and Platinum Elites (50K and 75K tiers) can only upgrade companions within two hours of departure (i.e., an unlikely upgrade). Silver Elites (25K) can not upgrade companions. (NW is probably the same).

DL: Companions can not be upgraded using the upgrade segment point. However, Medallions earn North American upgrades every 20,000 miles, and Platinum Medallions (100K) additionally receive six worldwide upgrades. These can be used for companions. On the other hand, they are fare restricted (same as if used by elite member).

Just another reason US Airways has the best upgrade program on the planet!