If Emperor Ashoka didn't have a change of heart after the battle of Kalinga and had not adopted Buddhism, that religion might have wound up as Jainism, a similar Sramanik religion to be restricted to the borders of India. Asoka's grandfather Chandragupta Maurya had adopted Jainism after leaving the throne. He became a Jain monk and starved to death as per the practice among the Jains. But Chandragupta adopted Jainism after he left the throne and hence the religion did not get much benefit out of his action. One weakness of Jainism over Buddhism might have been their rigidity in the principle of nonviolence which is extreme. Buddha always emphasized on moderation in all matters. Chandragupta's successor and the father of Ashoka, Bindusara is said to have favored Ajivikas, another cult similar to the Buddhism and the Jainism, but that cult disappeared like many others.
Ashoka spared no effort to spread the message of Lord Buddha to all the lands where he could and enabled the growth of Buddhism in East and Southeast Asia. Ashoka also served as a model for later kings like Kanishka. Today Buddhism is a major religion in Southeast and East Asia. In its absence, it is difficult to state which religion would have taken its place. In the last few centuries Islam or Christianity could have spread in Southeast Asia. In China and Japan, native religions like Confucianism, Daoism, Shintoism etc. were already there. The Confucianism also could have some influence in the Southeast Asia.