Even if mainframe computers hadn't been available, it would have been done like the Manhattan Project was, by many human computers at electro-mechanical calculators. Slide rules are to imprecise for actual orbital calculations.
After the war, though, some of the calculations for the "super" - the Hydrogen Bomb - were done on electronic machines. In particular, there was a concern that the Super might ignite the atmosphere and the calculations to check that were a bit much for rooms full of people coordinated by pneumatic tubes.