144 hour visa

Unfortunately, you cannot enjoy 144 hour visa free transit because you are not eligible with your itinerary for any visa free transit option. The reason is simple... You are not allowed to have two consecutive stops within mainland China within the period that is longer than 24 hours. If you only had just one destination within mainland China, like:

UK - Shanghai - Tokyo

You would be eligible for 144 hour visa free transit. Of course, a more complex itinerary would allow you to visit two mainland Chinese destinations, like this one:

UK - Shanghai - Hong Kong (or Macau)
Hong Kong (or Macau) - Beijing - Tokyo

This would imply one flight more than you have planned, but this itinerary would cover your both mainland Chinese destinations, each with 144 hour visa free transit.