Obviously nobody actually used the English word "empire."The difficulty here though is that there is no direct Latin translation for "empire," despite the fact that the English word originates from one of the most quintessentially Roman words, imperium.

The word comes from the verb impero, "I command." Imperium itself appears very early in the Latin literary lexicon; the playwright Plautus uses it to indicate a command:

Ámphitruo, assum. sí quid opus est, impera, imperium éxequar. "Amphytron, I am here. If there is any work, command me; I will follow your order [imperium]."

and the power/right to give commands:

Jubeo hercle, si quid imperii est in te mihi "I do indeed command you, if I do have the right to do so [imperii] to you."

These are its two most basic meanings.

Importantly, the word implies a bestowed power, not an innate or inherited one. The kings of Rome held imperium, acknowledging the unspoken contract to rule absolutely. It could also be used of various officials in the later Repulic and into the Empire; a provincial governor was legally bestowed imperium and it could be just as legally revoked. By extension, imperium could refer to region over which an official exerted control, or even the entirety of the Roman lands. However, it's use was not limited to what we consider the Roman "Empire." Cicero, during the late Republic, often used it to refer to all of Rome:

finium imperii nostri propagatio "an expansion of our realm's boundaries"

And while use of imperium does increase after Augustus' reign, there is no demarcation of "here is the imperium, before it was not."

Why does it increase then? Well, the related word imperator referred to the top military commander up until the first few decades A.D., and imperium often meant specifically his authority. You may recognize this word as the source of our "emperor;" and it was indeed one title claimed by the emperor. By the end of the Julio-Claudian dynasty, it was almost exclusively reserved for the Caesar. Thus, the unspecified, concrete imperium more frequently referred to the imperator's imperium. Many political crises began when the legions bestowed imperium on a favored individual. But as before, the imperator title is not inherently associated with the Roman Empire. Augustus, upon consolidating the offices of many former officials, took the title princeps for his newly created position. If there is any word that signaled the new era of Rome, it was this.

TL;DR: The best English equivalent for "empire" saw growth during the first few decades after Augustus until it took on its full imperial implications. However, it was used before Augustus with more or less the same meaning, and its use afterwards was still as a concrete extension of the abstract authority of imperium.

If you want to learn more, U Chicago's Logeion is a great resource for uses of Latin words