Can one take advantage of the scheme twice?

Joe, all the itineraries you have presented are eligible for 240 hour visa free transit. There is no doubt about that. Also, it is definitely and without any doubt permitted to have two 240 hour visa free transits within a very short time frame. I know you have written "New Delhi", although there are no yet direct passenger flights between China and India, although their re-start is in a plan. So, when you use 240 hour visa free transit, you will have to have a confirmed ticket that will take you out of China. You cannot have just a ticket that will bring you into China. Simply speaking, you will have to have a proof in your hand that you will be really in a transit by presenting a ticket from China to a third country or eligible territory as it is Taiwan for example. In any case, nobody will ask you for your return tickets or better to say for the tickets that you will use to enter (transit) China for a second time. This is a detail that does not interest anybody in reality. It will become a detail of interest at the moment when you perform your check-in in Taiwan for your flight toward mainland China and then toward a third country after mainland China. I hope this helps :).

Thanks for such a full answer. And impressive that you knew, or checked, about New Delhi. I had only used New Delhi as an example; my actual intent (that is, a hope for a possible trip) is to get to somewhere in India on that return journey.

A short follow-up question. If I take the now likely London - Shanghai - Taipei route, can I book a currently available flight that is listed as "1 stop" in Shanghai, or do I have to book two flights: London - Shanghai *and* another Shanghai - Taipei flight?

It is perfectly the same. It is allowed to travel on a single or on separate tickets. Both options are acceptable as long as your itinerary is within the 240 hour visa free transit set of rules. As you can see, your itinerary London - Shanghai - Taipei is eligible without any doubt and you can choose what type of tickets you will use. It is just important that your Shanghai to Taipei ticket is confirmed and paid and that you can present it in both London and in Shanghai. Have a nice journey.