I think your best shot is to wait for a "special occasion" when the risk of getting a disease because people don't wash their hands are at the highest (like when there is the flu running around).
First, choose a sign you would like to be put in the toilet. Signs are good, they are impersonal and a constant reminder. People won't get defensive over a sign but they might over a human. And if they feel accused, they probably won't see your point (that not washing your hands is dangerous).
When you choose the sign, be careful to choose one who explains why it is important to wash your hands. People don't neglect to wash their hands because they don't care, they think it is a stupid and useless rule which is only there to waste their time. I used to think that too. But if you put a sign that explains why washing their hands is important, they might change their mind.
So, for the sign, I will suggest something like that:
When you have your sign, go to HR, explain that there is a flu running around and that you are concerned that some people forget to wash their hands (this is important, you don't want to say that people do it in purpose, your HR might be one of the people who don't wash their hands) and are spreading the disease.
If HR answer with a categorical no, drop it. You don't want to be on bad terms with HR.
If HR agrees with you, then you can offer to take care of it. This way, you will be sure that the sign will be put in place within a short period of time.
If HR agrees and says they will take care of it but don't, you can try a slight followup. But don't be too insistent, you don't want to be on bad terms with HR.
A completely different approach would be to ask no one for anything and just put the sign in place (without anyone knowing that it's your doing). But be preprared for it to be removed (and maybe an angry sign/email from HR asking people to not put signs in the toilet).