It depends. Some units do provide quite a substantial boost to your entire team with their Awakened passives so you would want them up and running sooner. For example, Sherry gives your gold and below units +5% boost to attack, defense and HP, so a predominantly gold-silver team will become quite a bit stronger with her Awakening.
But generally, you may take your time with it, as training the rest of your team up to sniff will most likely give you more clearing power than Awakening one warrior.