I had a neural computation professor once that pointed out an excellent example of how "analog" techniques could be used on embarrassingly parallel problems to reduce the asymptotic bound of a computation:
Take a bundle of sticks of different sizes. There are many algorithmic ways to sort that bundle of sticks from longest to shortest with O(n*log(n)). An "analog" way to sort that bundle of sticks would be to stand them on end and let the sticks rest one end on a table (1 step). Now you have all the sticks with one end at the same level against the table. Take your hand and place it on top -- it will hit the longest, remove that stick and repeat for N steps. This process is O(N + 1) which is O(N). The key here was stacking the sticks on end -- a massively parallel solution to order the other ends of the sticks along the z-axis (up).
This is a neat thought experiment and can give an idea how analog solutions can reduce the asymptotic bound of an algorithm in a simple manner. Two huge caveats here:
1) We haven't taken an NP problem to a P problem with this example (more on that later) and
2) if you used N processors to sort N items you could sort the numbers in O(log n) time (with a big constant), so the reduction is not magical. Sometimes the analog resources needed solve a problem in a highly parallel manner are cheap. Another example of a cheap resource would be neurons (biological) for complex learning and pattern recognition.
Neurons can also put the apparent NP => P into perspective. NP problems are NP to find the optimal solution. You can find a "good enough" solution in P time that will work fine in nature. Evolution selects for highly efficient solutions that are "good enough". Think about how good the average person is at identifying objects in almost O(1) time. That's because there is a lot of parallel processing going on, and your brain still doesn't always come up with the optimal solution. For example, optical illusions, or forgetting where you put your keys (which would easily be O(1) for a computer!).
Another point with NP vs P in nature: solving NP to find the optimal solution is not the same as identifying the optimal solution. Identification of an optimal solution to an NP problem can be done in P time. Again recognition of a "good enough" solution will work rather than an optimal solution. Take the example of protein folding -- this is an example of nature doing all of the above. It takes advantage of molecular interaction forces that all work in parallel (no need for the natural folding "algorithm" to address one atom at a time as a computational algorithm does). Also, there is no guarantee that the (functionally) optimal solution to the protein folding will be found.
There are many examples of diseases due to protein misfolding. As @PeterShor pointed out sometimes the "natural" algorithm doesn't work at all (leading to a thermodynamically optimal solution, but not a functional one). That is where chaperone proteins come in -- they guide the folding into the correct functional form (a thermodynamic local minimum). Correctly formed proteins also interact with other proteins for transport to the right location so the "bad" ones (where the heuristic algorithm didn't actually solve the NP problem) are often degraded without being transported anywhere. All of these transports and folding mechanisms are happening with massive parallel pipes. Multiple transcription and processing mechanisms are simultaneously converting DNA->RNA->Protein at different points on a gene sequence. Every cell in your body is doing the same (but with different chemical messages about what to produce).
So, in short: How does nature do it? Tricks and Parallelism. Generally it's not actually turning an NP problem into P, it's just making it look easy.