Clearly, washing hands frequently with soap and water and using Purell will become the norm.
What will happen on Tisha Be’Av and Yom Kippur?
The halacha is really going to stay the same. The halacha is that on these days we don’t wash ourselves for enjoyment, but when something has to be washed to remove dirt or for a medical reason it is permitted. What will be needed in order to keep hygiene in order to prevent spreading disease, will be permitted, but anything in excess of that will be the same as usual.
O:CH 554-9, 613-3, M:B 554-19.