I was having this conversation with another scientist earlier today, and we realized we didn't know the answer. In short, by now pretty much everyone in science accepts the Big Bang theory as the way the universe began, and that it currently appears the universe is likely to keep expanding forever, but these are all things that basically come about from Einstein's relativity. But before relativity there surely were scientists who were trying to tackle these questions without relying on religion, right? I mean the 1800s gave us the realization of things like how the universe must be far older than a mere 6,000 years thanks to geology, so surely people then started to speculate on non-religious ways for the universe to begin that didn't require a Creator, and ways it could end that weren't Armageddon.
Thanks!
Most theories previous to Big Bang are of an infinite spatial and temporal universe, and fall under the category of Static Universe theories. This actually started back in the 16th century with philosophical ponderings and was the dominant opinion until the early 20th century.
Einstein's relativity didn't help explain the Big Bang right away. As a matter of fact, Einstein proposed a modified form of Static model based on general relativity that was temporally infinite but spatially bounded. It was Hubble's Red Shift that launched the Big Bang, along with the observations of the trajectories of the red shift (Hubble Flow). Einstein went along with the new Big Bang, and of course relativity does apply to the Big Bang universe, so Einstein's universe model was wrong, but his relativity theories were right. Later on Georges Lemaître uses relativity to discover the Hubble Constant, and solidify the Big Bang theory.
Looking back farther into the 19th century, you see some prominent scientists who are religious, but they are agnostic and not following a strict interpretation of their religion. Lord Kelvin saw a static universe that would wind down, according to his laws of thermodynamics, and become cold dead matter. Being religious, he saw the beginning of the universe as created by god. On the other hand, the respected astronomer Richard Proctor describes a universe with no beginning and no ending, in both time and space, in his book "Other Worlds Than Ours", 1877. A classical static model that is common to the time. EDIT: Proctor also believed in god, but saw god's role as "supervision and control" of the universe.
So even though some of these people were religious and some were not, they often shared the same views of the nature of the universe.