Namo tassa is not itself an isolated later composition; it is a formula embedded throughout the Pali canon, opening numerous discourses in more or less this wording, and the canon's closure by around the first century BCE gives an outer limit for when its phrasing was fixed, though the formula's age within the oral transmission that preceded that closure is earlier still and cannot be dated more precisely. Its transmission since has been among the most stable in Theravada Buddhism of anything recorded in this atlas: it is said three times before the going for refuge and the taking of precepts and before essentially any act of formal chanting, across Sri Lanka, Burma, Thailand, Cambodia and Laos alike, with negligible variation in wording despite those countries' very different vernacular languages, because the Pali is preserved as Pali rather than translated into the local tongue for ritual use.
The three epithets it applies to the Buddha, bhagavato, arahato and samma sambuddhassa, are technical terms whose English translation is debated among Pali scholars rather than settled. Arahat is variously rendered worthy one, noble one or, following older translation habits, saint, each carrying different connotations the Pali term does not straightforwardly have. Samma sambuddhassa is harder still, compounding samma, right or complete, with sambuddha, awakened, itself intensified by the reflexive sense of self awakening rather than awakening taught by another, so that renderings range from perfectly enlightened to supremely self awakened, and translators including Bhikkhu Bodhi have written on why no single English phrase captures both the completeness and the self attained quality the Pali compound asserts together.