144 hour AFTER 72 hour visa waiver

Joeyc, no, you cannot do that. As for the 72 hour policy, you can only transit in one Chinese city. So after staying in Beijing, you should fly to a third country or region, actually, you can fly to HK directly. But anyway, you cannot fly to Shanghai.
If you want to enjoy the 144 hour free transit in Shanghai, you can first fly to HK from Beijing, and then fly to Shanghai from HK. After transiting in Shanghai for 144 hours, fly back to UK directly.