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How to deal with too high load

I am afraid of the following scenario: A web application using plotly dash experiences a sudden high load due to many users, which is too high for the servers serving the app.
In my case, clients contact an nginx instance which serves static content itself and refers to workers for the computation tasks. These tasks will not have the required resources when demand is too high.
In my tests, this leads to either no response after the initial static page is served by nginx, or some error message like “Error library cannot be loaded”. (Server response to client is 502 to serveral requests)
So my question is: How to deal with this situation. I would like a solution, that allows users to show a decent message like “Due to high demand the application is not available. Please try again later.” Without having the app look ‘broken’.
Any ideas appreciated. Thanks!