Far East tour

You can leave the airport as long as you get a 24-hour visa-free entry permit at the border inspection counter. You don't need a visa for Taiwan and you can stay in Taiwan for up to 90 days holding a UK passport. Taiwan doesn't issue the 144.
If I understand right your route is like this: Taiwan - Shanghai - Seoul - Tokyo - UK. You are eligible for the 144-hour visa-free transit in Shanghai so you don't need a visa for Shanghai if you plan to stay there less than 6 days.

Your first stop in China, in Shenzhen city, will be covered by 24 hour visa free transit and you do not need a visa. Also, you do not need a visa for Taiwan because Taiwan allows UK citizens to enter without a visa. Your passport is enough. So, your visit to Shenzhen is covered by 24 hour visa free transit and your visit to Taiwan is absolutely visa free. When UK citizens visit Taiwan, this is the same as if they were visiting France.

When you arrive in Shenzhen, you will first approach transit counter where you will apply for a temporary entry permit in order to be allowed to exit the airport. Here you will present your flight tickets and for this reason you are strongly advised to have all your flight tickets printed out and at hand, ready to be presented to the airport authorities. Also, if you plan to have a decent overnight in Shenzhen, you will book a hotel there, so in this case you are advised to print out your accommodation booking confirmation. This application for a temporary entry permit is free of charge. When a permit is granted, you will cross the immigration and spend your layover time in the city.

During your second arrival in China, in Shanghai city, you will do the same procedure, but here your stay will be covered by 144 hour visa free transit since your entire transit time will be longer than 24 hours. You do not need any kind of visa. During your stay in Shanghai you are free to explore Shanghai city and two additional provinces, Jiangsu and Zhejiang. In short, you are good to go. Please, here just one warning... You are not allowed to have any additional landing within mainland China on your route from Japan / South Korea to UK via Shanghai. Have a nice journey.