No. Not even close. Not even a little tiny bit. On a human scale a nuclear explosion looks like a huge event of immense power, but from the Earth's point of view it's the tiniest of unnoticeable pinpricks. The Earth is a huge body and the amount of energy it would take to alter its axial tilt in a single shove is more than humanity has ever produced in all of its history put together. If we tried to do it on purpose and detonated all of the 10,000 or so nukes the Russians and the U.S. own all at once, it still wouldn't be enough to nudge a body as massive as Earth. And actually changing Earth's orbit so it was heading for the sun would take many orders of magnitude more energy. That being said I will point out that the movie does a good job of showing the kinds of climate shifts you'd get if an event like this did occur. The writers were much better climatologists than physicists. But if the Earth's axis shifted by 11 degrees as they describe it would be immediately obvious to even amateur astronomers, because constellations from the southern hemisphere would suddenly be visible. There would be no denying it.

Keep this in mind when you judge this era.

The Soviet Union took the cold war very seriously and made a strong effort to infiltrate our government. The few who knew about Venona back then couldn't mention in public how they knew that there were communist agents (that would have compromised the operation), which made them seem to the public to be overly paranoid. They were not.

"In some cases, as people were looking on each other and talking, they began to totter and fell on the streets and at home. It might happen that a person was sitting at work on his craft, holding his tools in his hands and working and would totter to the side and his soul would escape."

He goes on to describe how everywhere one looked were:

"corpses which split open and rotted on the streets with nobody to bury them."

This plague killed more than a third of the population of Europe and Asia. My point is that pandemic diseases are not unknown in human history. They've happened before and will happen again. But a disease that turns you into a vampire? Unlikely, I'll admit, but I have no problem with it here. You don't become a superhuman immortal demigod like Dracula. Instead you become a deranged shambling pitiful creature that feeds on blood.

Of all the reasons why they should have known where they were, the most obvious is life.

Every person, every oak tree, every blade of grass should have been a clue. And let's not even get into the fact that the apes spoke perfect, 20th century English! At the very least they could have made mention of the idea that perhaps after they left Earth some one invented faster than light space travel and colonized this world a thousand years before they got there.