How much does COVID-19 affect weather forecasts?
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Given that the crisis is still ongoing, I doubt we know the full extent of the impact (forecasting the weather is hard enough, forecasting the impact of a pandemic is outside my expertise). But, at least one conference  was supposed t
o go to was cancelled. And I know the lack of airplane obs is impacting data assimilation. Beyond that, I am not sure. Some hypotheses perhaps include:


Is the change in the quality of life may be impacting forecaster accuracy?
What impact does the delay in communication of scientific research have on forecast accuracy?
How does this impact the future science budget?
How does the absence of airplane data degrade forecasts (could be evaluated using a OSSE experiment)? On a related note, how 
How is the communication of weather forecasts by in-studio meteorologists compare to meteorologists that are in self-quarantine?


How this ordeal has affected weather forecasts may be a good cross-disciplinary study for the future.

To fully answer your question, you can look at Benjamin et al. (2006) to evaluate the relative impacts of different observational systems. Certainly there have been some changes since then, but this can give a cursory idea of what we might be looking at.