There is an interesting contemporary Roman literary source for this. Petronius, the master of revels for Nero, wrote "The Satyricon," sometimes considered the first novel. Among other stories and incidents, Petronius describes a Roman soldier who was detailed to guard the corpse of a crucified criminal, to keep the family from taking the body and burying it. The widow of another dead man (the place of crucifixion is a cemetery) loudly mourns his death nearby until the soldier convinces her to live and seduces her. They enjoy each other's company until the family of the dead criminal notice the watch is loosely kept and steal his body away. The soldier is ready to kill himself since the penalty for shirking his duty is death. His new lover practically points out that her dead husband's body will do just as well. They mount it in his place and the neighbors wondered how a dead man managed to climb the cross.
The story is seen as an early slam at Christians. I think it is proof of how even by Nero's time, the facts of Jesus' crucifixion and resurrection were already well known at Rome. The story is also a good source of crucification information.
Gutenberg text of "The Satyricon" is at http://onlinebooks.library.upenn.edu/webbin/gutbook/lookup?num=5611
Eusebius writes that Tiberius heard of Jesus' reputation for miracles, and put Jesus' name before the Roman Senate, asking them to declare Jesus one of the Roman gods. (Miracle workers of that sort were a far more accepted phenomenon in those times than they are now.)
The Senate did not confirm Jesus.
But what if they had? Jesus would simply have become another member of the Roman pantheon... (I could go on at length with speculations, but you can paint your own alternative history...)