Rotkreuz nurses were issued with greatcoats with the red cross for collar tabs, so I think NH would also get them, it's not like they were sent home in the winter anyway. And if not issued, I'm sure no one would beat them for picking one up and using it, I'm sure a gal could seduce a quartermaster

As for helmet, yes, they wore them, well kept them nearby, I don't think they were officially issued but their workplace would keep spares in case of an air raid. Same as the ladies in England, except they carried their helmets everywhere.