I haven't played Nayuta yet (got the GOG release a few days ago though) but I've played all of Cold Steel I~IV and did not notice anything in them that would suggest that Nayuta even existed -- unlike Zero/Azure where not having played those (due to no official English release at the time) led to some "who the heck is this and WTF is going on here?!" moments as well as some "I would probably appreciate this more if I'd been able to play those games..." sections.

If you want to carry on into Cold Steel after Azure you'll be fine.