Flight Itinerary.

Unfortunately, this is not possible. Simply speaking, under 144 hour visa free transit scheme you are not allowed to have two mainland Chinese cities in a row. You can visit just one transit zone at the time and the immediate points before and after it have to be in two different countries or eligible territories (eligible territories: Hong Kong, Taiwan and Macau).

To conclude, you can fly:

London - Shanghai - Hong Kong

but your original itinerary is not eligible and if you proceed with it, you will need to obtain a Chinese visa in advance.

I hope you will be able to adapt your itinerary... For example, you could do something like this:

London - Shanghai - Hong Kong
Hong Kong - Beijing - London

This would be ideal combination and you would be eligible for 144 hour visa free transit twice.