The oldest bows we have found in Europe are about 10,000 years old. There are older possible bow fragments, about 12,000 years old. These are actually the oldest bows that have been found anywhere; boggy northern Europe was friendly towards preserving wood for so long. (Archery might be much older than this, possibly 60,000 years old.)

This is well before the Romans. The Romans used archers, both mounted archers and infantry archers. The main Roman military bow was a composite bow, essentially the same as the Asian horn-wood-sinew composite bow.

For more:

u/PapiriusCursor discusses Roman archery in https://www.reddit.com/r/AskHistorians/comments/54tra0/why_didnt_the_romans_use_archers/

I discuss the invention of the bow in https://www.reddit.com/r/AskHistorians/comments/9pgvu3/where_did_the_bow_and_arrow_originate_and_how_did/

For more on the finds of early European bows (mesolithic bows from Denmark and nearby), see Jan H. Sachers, "Mesolithic Bows from Denmark and Northern Europe", pp. 155-180 in Lee, Noh-Shin (Ed.), Study of Structures, Materials & Manufacturing Processes of World Traditional Bows & Arrows, Cheonan, 2009: https://www.academia.edu/11765815/Mesolithic_Bows_from_Denmark_and_Northern_Europe