In Ohio an RN can not challenge the paramedic exam (Ohio uses National Registry). Ohio does permit an RN who is an EMT-B to take a RN to paramedic bridge course, typically 3 months vs the 12 for full medic school, and then take National Registry.
A caveat from experience as a medic and paramedic instructor: nursing may or may not prepare a person well for being a paramedic. Nursing is very broad and if a person's experience isn't in something like emergency medicine, it won't prepare them well for being a medic. Part of it is specific medical knowledge, like cardiology and pulmonology. A much bigger part of it is the mindset difference. Most nurses, esp. in hospitals, operate in a very controlled, process driven environment. They have to focus very hard on providing care according to the facility's processes. Pre-hospital care is all about adapting to uncontrolled settings and situations and making decisions driven mostly by patient assessment and little history. The EMS crew is "it" and within the confines of their scope of practice have to be pretty creative. The RN's I've taught often take a fair while to adapt to this different approach. Medics usually take a fair while to adapt to hospital work, because their mindset is more focused on just getting the task done. It's a question of adaptation to different environments.