One of the things I remember quite vividly from my Psychology classes in college was the idea of a "satisficing" problem solver. Given a (reasonably solvable) problem to solve most people can come up with a satisfactory solution. The difficulty comes when you ask them to come up with a new solution. Many people struggle because their brains say "I already came up with a solution, and it was a pretty darn good solution too."
The really creative people are the ones who insist that their brains come up with another solution, and another one until they can be confident that they've found the best solution within a reasonable time investment.
To me this seemed to get easier as I encountered more "languages of the mind", i.e. more ways to think about things. This, is very much tied to the "nurture" part of our lives, as foundational experiences are more by the very definition of "experience" subjective and unique.
Solving a problem within mathematics in a new way can be made much easier if you have grasped multiple fields (for example, algebra vs. geometry). I've seen people understanding chemistry well because they enjoyed cooking, and could sort of use either to get to a given explanation to solve a problem. I've definitely started to grasp chemistry only when I reached a decent level in theoretical physics.
Here's my assumption: everything has a likelihood of depending to some degree on other things (examples: can you do mathematics without a language or writing? can you do physics without mathematics?).
Therefore, "thinking laterally" could very well be thought of more as "thinking with an interesting combination of previous vectors of thought". Perhaps the "genius" is to create nonlinear combinations of previous vectors.
So in short, this ability to come up with another solution, and another, and another.. I wonder how much it is tied to the richness of experiences you've had since your birth, and particularly the foundational ones (at the very least I would assume to be more testable than later on, if my hypothesis about dependency of "thought vectors" is true).
The really creative people are the ones who insist that their brains come up with another solution, and another one until they can be confident that they've found the best solution within a reasonable time investment.