What proportion of papers submitted to Nature are actually sent for review?
Edited: I found a better answer to the question, the original indicative answer is left below.

Here, we have a statement from December 2006 regarding the number of submissions, rate of editor rejections and ultimate acceptance number. While slightly outdated, the acceptance rate of Nature journal does not seem to fluctuate widely between 2006 and 2012 (always around 7-8% of submitted articles), hence I believe the number is significant and valid:


  Nature receives approximately 10,000 papers every year and our editors reject about 60% of them without review. ... ... In the end we publish about 7% of our submissions.


From that we have, that in 2006 about 40% of submissions were sent out for peer-review.


The original indicative answer:
Not a precise answer, but can be indicative also for Nature journal itself: here is a statistics of a decade 2002-2012 of publishing in Nature Materials in numbers, various stats are presented. Specifically this graph shows that the ratio of peer-reviewed vs. submitted manuscripts is floating around 13%. Out of peer-reviewed, about 60% get accepted. This shows that their first filter, the editorial decision whether to review at all, is extremely aggressive and in fact is the filter in the publishing process.

According to a note here, Nature Neuroscience sends about 30-35% papers for peer-review.