We don't know much about neutron stars and specifically about the interaction of neutron stars with other neutron stars or celestial bodies, infact there is a whole problem in Physics related to this topic called the Mass Gap Problem, so any answer here is just a speculation based on our current understanding of Physics, which is quite limited when it comes to exotic things like Neutron Stars, black holes etc.
The short answer is, we don't really know what would happen, we can speculate but that's about it.
I assume, nothing would happen, the two would continue to orbit each other just as they did before, just as if our sun turned into a black hole, all the planets would still continue to orbit it just like they do now.
But if they had an unstable orbit, I would assume the two would catastopically collapse ending up with a larger neutron star or maybe a black hole. This is where the mass gap problem arises where we don't know if there is something inbetween a black hole and a neutron star.