It is hypothesized that the Celtic and Italic (including Latin) languages split later than any other major subfamily of the Indo-European languages, but evidence is scanty. Probably proto-Celtic was spoken north of the Alps and proto-Italic south of the Alps sometime in the second millennium BC.

By the time the Roman republic replaced the kingdom in the 6th century BC, it is likely that the Celtic spoken in Britain had already split into the two branches called Q-Celtic and P-Celtic. The latter included Brittonic.

By the end of the Roman empire Primitive Irish and other Goidelic languages were being spoken in Ireland and parts of Scotland, along with Brittonic in Southern Britain.

So there were changes happening to the languages throughout the period of the Roman empire. But I don't think the phonological changes would have affected the word for Roman very much. The Old Irish word for Roman was Rómánach. We don't have comparable early literature for the other Celtic languages, but the word for Roman would have been similar. Even the current Welsh word is Rhufeinig, which is simlar to the current Irish word Rómhánach, where the mh represents an v sound.