Julian M. answered 02/05/23
Interdisciplinary PhD includes Comparative religion & mysticism
Hinduism and Buddhism have a great deal in common and share many ancestral roots. In India today, there is still a lot of ongoing conversation between the tenets of both traditions, and there is a lot of crossover between adherents of both. I have met many Hindus who admire and study Buddhism, and many Buddhists also know quite a bit about Hinduism and often consider it a cousin or predecessor. This is historically correct, as Buddhism originally grew out of Hindu Brahmanic heritage.
The specifics of your question would ask for one to get a lot more creative, and one could take the imagination of this in many different directions. In many respects, current New Age spiritulaities in some cases do exactly this - although they typically borrow from many other traditions as well.
If I were to play with such an integration myself, I would personally lean toward the evolution of consciousness and the inner work of liberation as my central theme. Both Hinduism and Buddhism carry strong threads of this. In Hinduism, we could call it Moksha and note that it is the goal of Yogic self-cultivation across many lifetimes. In Buddhism, it is Nirvana and is also the goal of the Dharma. In both cases, if one looks closely, one finds detailed (and much overlapping maps) of the stages that the soul goes through in this awakening journey of liberation, and the practices that are helpful to the soul at different stages. In many cases, these practices also overlap.
There are certainly differences, and even arguments, between the two religions. But if we focus on the practical (you might say, psychospiritual) aspects, then we find that both traditions are detailed systems of what you might call the science of soul development. If I were to imagine a new religion integrating the two, it would be based on the integration of the best practices and most sophisticated understandings of this inner science of self-evolution.