Final answer:
Buddhism significantly declined in India after a millennium, overshadowed by the spread of Islam and the evolution of Hindu practices.
Step-by-step explanation:
The religious tradition that mostly disappeared from India after a millennium, while Islam was competing and until it became a major religion especially in the northern parts of the sub-continent, is Buddhism. Buddhism saw a significant decline in India by the thirteenth century, as it partly fell out of favor amidst the rise of both Islam and the evolution of Hinduism into a form of worship that allowed personal devotion to individual gods. Despite this decline in India, Buddhism found refuge and continued to thrive in other parts of Asia such as Southeast Asia, Nepal, or Tibet where it remains an influential religion. By contrast, Zoroastrianism also declined but due to the Muslim intolerance for their distinct belief system, many Zoroastrians migrated to India, forming the Parsee community.
Meanwhile, Hinduism retained its dominance in the southern regions of India, largely due to distance from the Islamic centers. Sikhism, a later religion, has continued to exist and maintain its identity in India up to the present day. Hence, Buddhism is the religious tradition that disappeared from India after a millennium, competing with Islam.