Theravada was not the most widespread school at the time, nor were its doctrines yet systematised and formulated into the Vinaya and Abhidhamma the way it's read and practiced today until Buddhaghosa started his work on the Visuddhimagga... his teachings spread and took root in Sri Lanka and were spread to Myanmar, Thailand and parts of South East Asia including parts of Cambodia and Laos.
Theravada as a word itself is relatively new, as the schools at the time had derived from either the Sthavira or the Mahasamgha (Mahasamghika) traditions. Mahayana itself had its roots in Mahasamghika which had started evolving into Kukkutika and its companions and descendents, while the predecessor to Theravada itself was the Sthavira school which then evolved into the Sarvastivada, Puggalavada, Vibhajjavada and its descentants, one of which later evolved into Theravada.
At the time the more dominant schools of thought were the predecessors of what is now the Mahāyāna schools, some of which evolved later into the Tibetan Vajrayāna schools, and some went through the north west of India, through Gandhara and the Kusan empires into Central Asia, into China, Korea, Japan, Vietnam and parts of Cambodia.
There are several academic works that clearly lay out the history and evolution of the Buddhist schools from verifiable sources. I'll update this if and when I get the time to look into my bookshelf later :)
However you are completely right about Buddha Vandanā. This is not worship. In fact there's no worship in the traditional Pāli canon, or the Tripitaka, and in fact Sakhyamuni himself advocated against forms of worship. In Buddhist schools the Vandanā and other forms of such processes are for showing respect to the Tri-Ratna, i.e., the teacher, the dhamma and the sangha, as a means to bring oneself into a conducive frame of mind, to develop the mind.