Do I qualify?London (+ HK stopover) > Seoul > Shanghai (+ HK stopover) > London

You will be fine to use the 144 hr free transit in Shanghai by traveling like London (+ HK stopover) > Seoul > Shanghai (+ HK stopover) > London. Nice trip.

Thanks Megan! I emailed the Chinese embassy in London to check and they said they dont have any information on the 144 h free transit. They replied:

Q: What about the 144 hour transit policy/ What about another transit policy?
A: We have no information about the 144 hour transit policy or other transit policies not mentioned on our website.

I'm assuming its ok as others have done it but weird they act like it doesnt exist.

If worst comes to worst, can you get a visa on arrival on the airport?

Embassies and consulates are not responsible for entry rules. To get official confirmation, one has to call the immigration office at the designated port of entry during regular business hours or use Google translate to check out their Chinese-language website.