Well there is no real way for Nagato to awaken the Rinnegan or have it technically unless they are implanted. The story goes that the elder son was gifted with the sages eyes and the younger son the sages body, the Uzumaki are related to the Senju. This means that Nagato would have inherited the sages body(not entirely but pieces of that power, kinda like how Naruto has massive chakara reserves). The Rinnegan is awoken through the Uchiha clan. The body in a sense is awoken through the Senju. I guess it all stems down to reincarnations of the elder and younger son of the So6p. Which obviously Madara and Hashirama were that.
Even if he is Madara's son, Nagato can't have both traits from both clan. I do remember that being stated in the manga. You can only have one trait. If that was the case, you would have had some insane Uchiha's back then.