I haven't worked out the details (which will be very complex, probably involving serious supercomputer resources to model) but there will be gravitational perturbations of each orbit by the other bodies. Here in the solar system, the existence and position of Pluto was deduced and calculated by very careful observation of the orbits of Uranus and Neptune. Once they knew where to look, they pointed a powerful telescope at that patch of sky, and there was Pluto.
For this system we can test GR rather than simple Newtonian gravity, because the bodies are very dense and close. It tests the equivalence princible, because a Neutron star is made of Neutrons and a White Dwarf is made of ordinary atomic nuclei (ie with protons, and electrons between the nuclei). Short of a black hole, that's about as much different as two astronomical masses can be. (*)
I'll have to file this system along with Zircons dating of the age of the Earth. $DEITY is being extremely kind to us, providing us with the means to scientifically investigate questions we might have thought were forever beyond our grasp.
(*) I've always wondered whether bosons (commonly neutrons and protons) and fermions (commonly electrons) are gravitationally equivalent. Is there any way to probe this, even conceptually, given that the electromagnetic interactions of electrical charges are about 10^40 times greater than any gravitational effects?