144 hours transit visa

No, unfortunately, you don't.

As you can see, your first itinerary is:

London - Beijing - Shanghai - Taipei.

Having two consecutive stops within mainland China immediately limits you to 24 hour visa free transit and if you cannot perform your itinerary within 24 hours, as is your case, you need a Chinese visa obtained in advance.

Your second itinerary:

Taipei - Beijing - London

Is definitely an example of itinerary that is eligible for 144 hour visa free transit.

Now, you have to decide what will you do. You can rearrange your flights and to be eligible both ways or to obtain a Chinese visa if you want to retain your actual itinerary.