At a railway station in central Sydney, I saw a door marked EMERGENCY EGRESS ONLY. I guess that at least 99% of such doors in the English-speaking world are marked EMERGENCY EXIT ONLY.
Egress is the slightly earlier word, dating from the 1530s. Exit as a stage direction (technically, a verb) as in ‘Exit, pursued by a bear’ dates from the 1530s, but from the 1590s it was used as a noun, as in ‘All the world’s a stage, And all the men and women merely players; They have their exits and their entrances’ and (occasionally) a ‘real’ verb.
From about 1650 to about 1850, the two words were used more or less interchangeably, but then the use of exit grew and egress declined, probably corresponding to the growth in public railway travel. Then in the early 1970s, the use of exit skyrocketed, for reasons I can’t think of, but curiously declined from 2000 to 2008. Most of this was due to the use of exit as a noun; exit really only began to be used as a verb in the 20th century.