CanadianGran
I don't feel sorry for 'the poor woman'. The OP suspected she is not washing her hands, paid attention to noises and confirmed her suspicions. This is a charity job and most likely does not have a safety inspector, etc.
I feel we cannot be responsible for others actions, and can only be responsible for ourselves. You can hint about the taps taking too long to get warm, or the scent of the soap and perhaps she will get the hint. In the meantime, you are washing your hands, and make sure to grab a tissue or paper towel to open the door on the way out. Then your hands remain clean.
You don't say how physically closely you work with this woman. Do you share a desk area, or computer etc? Is there any way that you don't have to touch what she touches? As long as you don't have to touch joint surfaces with her, then you should be fine.
Good grief, ‘you don’t have to touch what she touches’, the poor woman is not a leper. How spiteful and nasty. I notice this is a charity that the OP works for and the woman who is being ‘bullied‘ (for that is what it is, whispering about her and setting traps) is a volunteer, instead of someone having a word with her about hand washing someone should warn her about the nasty colleagues she works with.