Hi, I am flying into Shangai and then onto Xian for 2 nights and then onto Beijing for 3 nights.

Actually, the full visa is a must because your first visit to China simply cannot be treated as a transit since you will be traveling through three different transit zones consecutively and that is not allowed. Totally different is your second visit to China that is definitely eligible for 24 or 144 hours visa free transit. In case that your itinerary in Shanghai is shorter than 24 hours, then it would be 24 hours visa free transit and in case that your stay there is just a minute more than 24 hours, you are eligible for 144 hours visa free transit. To conclude, UK citizens are qualified for multiple-entry, long term visa, valid from 2 up to 10 years. You should keep in mind that you have to ask for such a visa because otherwise they will issue just a single entry visa.