Taipei>Hainan>Shanghai>London

Part one:

Andy, you are not allowed to do that under Hainan 30 day visa free access. You are allowed to enter, stay and exit Haian under the mentioned visa free option, but you have to arrive in Hainan from outside of mainland China (arrival from Taipei is fine) and also leave Hainan to the country different than mainland China.

I assume that you have probably thinking about Hainan VOA that allows you to perform, exactly what you have mentioned:

Taipei - Hainan - Shanghai - London.

This is perfectly allowed under Hainan VOA, but you should keep in mind two extremely important details:

1. UK citizens have to pay for Hainan VOA 304 CNY upon arrival
2. The largest majority of airport staff in different countries and territories are reluctant to allow boarding to passengers bound for Haikou or Sanya (two airports in Hainan) if they declare that they will rely on Hainan VOA. You should be lucky to find well informed member of the airport staff who will allow you to board the plane. Recent experience tells that the airport staff in Singapore has denied boarding to a group of UK citizens who have declared their intention to obtain a Hainan VOA. So far, most of those who have successfully traveled based on Hainan VOA have used Hong Kong because there the airport staff is really well informed and in any case they can easily contact their mainland Chinese colleagues. I would assume that the airport staff in Taipei is also very well informed, but I really cannot tell you that you should travel with absolute peace of mind because I'm afraid that you could encounter a member of the staff who could denied you boarding.

Part two:

Under the assumption that you are allowed boarding, you will easily obtain Hainan VOA by applying and paying the fee. After your stay in Hainan, you are free to take a direct domestic flight from Hainan to Shanghai where you can stay for some period, but this period is not 144 hour visa free transit, but a stay in China under the Hainan VOA rule. You are, of course, perfectly allowed to take a direct flight from Shanghai to some other country.

Now, you have to decide what to do because you are theoretically allowed to use Hainan VOA and to perform your initial itinerary, but in the same time you will have to take some risk to be denied boarding in Taipei.

Your second solution is to use 30 day visa free access and to enter Hainan from Taipei and after that to exit to, for example Hong Kong and from there to fly to Beijing, enjoy 144 hour visa free transit and fly from Beijing to London.

The third solution is to obtain a Chinese visa in advance and to perform your itinerary using a Chinese tourist (L) visa.

Now you know some of the solutions and you can make a decision.