24 h visa free rules with two transit stops in China

Your itinerary is definitely and without any doubt eligible for 24 hour visa free transit and you do not need any kind of visa, guaranteed.

You will have just one tiny formality to accomplish... Since your layover in Shanghai is a short one, the first thing you have to do is to approach transit counter (desk) where you will apply for a temporary entry permit to be allowed to formally enter China because your next flight is a domestic one. Here you will present your flight tickets and I suggest you to have them as printouts to speed up the process. This application is free of any charge. After a temporary entry permit is granted, you will proceed through immigration and after that you will run to the gate of your flight to Beijing. In Beijing you will cross the immigration because your flight is an international one and you will proceed to your gate for a flight to Warsaw.

As far as your last question is concerned, you have to know that this is not your problem because you have bought your tickets that allow you to enjoy 24 hour visa free transit. Every possible occurrence, being it a delay or a cancellation do not affect you and your rights to travel visa free because this is not the circumstance caused by your negligence or anything else... In other words, even if you will have to stay a day longer in China because of the airline fault or the weather conditions, you will be covered by this visa free transit scheme as if nothing has happened.

Dear Borut, thank you for detailed explanation. I have one more question - do new regulations allowing foreigners eligible for the 24-hour visa-free transit to be exempt from border inspection procedures change anything in our situation/ formalities at the airports?

No, your situation does not change at all. Everything I have written is valid and applicable in your case and you can freely proceed with your travel plans.