I don't know if it's just handwashing, but Australia is reporting a significant decrease in flu this season:
Following a high start to the 2020 interseasonal period, currently, influenza and influenza-like illness (ILI) activity are lower than average across all systems for this time of year. At the national level, notifications of laboratory-confirmed influenza have substantially decreased since mid-March and remain low.
In the detailed PDF for May, which was linked from the first link above but is now gone, they say:
Interpretation of 2020 influenza activity data should take into account, but are not limited to, the impact of social distancing measures, likely changes in health seeking behaviour of the community including access to alternative streams of acute respiratory infection specific health services, and focussed testing for COVID-19 response activities. Current COVID-19 related public health measures and the community’s adherence to public health messages are also likely having an effect on transmission of acute respiratory infections, including influenza.
And provide a lot of graphs. I think this one is the most impressive:
The "second half of June" detailed PDF is here, but may go away. Future readers should go to the first link and look for links under the Full Report heading.