It is difficult to say what the fastest-growing religion in the world is. How do you define that? Is it the religion with the most converts? The religion that is growing most, percentage-wise? The religion that is growing most in terms of sheer numbers of people due to birth?
Anyway, depending on how one measures "fastest-growing religion", one might get various answers, but Islam is often said to be the fastest-growing religion today. Perhaps, this is in part because another gigantic world religion, Christianity, is losing adherents in the modern era; although there is still active proselytization worldwide, and many people convert to Christianity, it doesn't have as strong of a foothold in some other areas as it did before.
In my view, the important thing, however, is quality not quantity. Allah doesn't need billions and billions of people to profess adherence to Islam in name only. What is important is the sincerity in looking to Islamic teachings and scriptures to build a world which facilititates human and spiritual growth for individuals and societies.
We don't know what the future of science will be. However, I think we are leaving the era of "scientism" (i.e. science as a substitute for religion) and moving back to what was the norm throughout much of human history which is a worldview containing both the material and the immaterial. As you mention, people do have an innate interest in the big questions of existence and that will not change. In particular, people tend to turn to the spiritual of religious in times of difficulty or crisis, when material solutions fail, and that also will not change.
However, whether that will correspond to a growth in Islam/other current religions or the development of new belief systems altogether is something that only time will tell. I think a large part of that depends on how Muslims present Islam (both to potential converts and to the younger generations). Although the Qur'an treats Islam as a theology not as a racial matter, there is a tendency today (among Muslims and others) to treat Islam as a racial matter and Muslims as a race or to treat Islam as the property of certain ethnic groups. So, this does not encourage the average person who is not a Muslim to consult Islamic scriptures and teachings for theological answers. Maybe it also depends on how Muslims respond to the challenges of the next centuries and what Islamic thought has to contribute with respect to these challenges. Islam has been shown to be persistent in the face of challenges so it will likely remain a significant force in the future.
Of course, after the appearance of the Mahdi (A), many people will choose to follow him because it will be apparent that he has the truth and what is good for people.