Can I demand my passport back in the middle of an application for a Chinese Visa in the UK?

It depends on when the summer school will actually be and up to when you need to attain the required visa. I can see one of the following options:

• Applying for the visa before you leave for Rome, hoping to get the passport back in time.
  • Attempt to pay more for quicker processing
  • If it doesn’t work, attempt to request it back
• Applying for the visa after you come back from Rome, hoping there is enough time until the school starts.
• Applying for a second passport. However, it will likely take much longer to get the second passport than the visa. Also here for completeness’ sake.
• For completeness: Having a national ID card to spare. Then hand in the passport and use the national ID card to travel to Rome.

The last option is only an option if your country of citizenship issues national ID cards; the United Kingdom is one of the few on the list of EU countries that don’t. The European Union’s web page confirms that you need either a national ID or a passport to cross a non-Schengen border.

Checking out the web page of the Chinese visa service centre, there appear to be three options: regular service, express service and postal service. Regular service is quoted at taking four working days, express service as three, so it appears you have already found the fastest possible option. It also costs £27 more than the regular service as per their schedule of fees (£178 versus £151). Since the difference between the two is only one working day, I don’t know if I would deem it worth the money.

What I have personally never heard of is demanding one’s passport back before the process is completed. There are a number of problems associated. For one the visa service centre is rather clear in that:

Clients acknowledge and hereby confirm that the Application Centre is not involved in the visa assessment and decision-making process. The Embassies and Consulates have the authority to decide whether a visa will be issued or not, the type of a visa to be issued as well as its validity, duration of stay and number of entries in accordance with China's relevant laws and regulations.

So the application centre is not involved in the actual granting of the visa, and your passport will not be present in the application centre if you attempt to get it back. Another aspect is the following:

Clients must make their visa applications at an appropriate time in accordance with their travel arrangement. Under no circumstances will the Application Centre be responsible or liable for any delay of travel arrangement as a result of a Client's inappropriate action in regard to the time of submitting the visa application or the visa assessment result of the Embassies and Consulates.

While this paragraph is probably targeted towards one’s travel to China, you can just as well read it as ‘make sure you plan your visa application so that you don’t need your passport during that time.’ Thus, I very much suspect that if you go to the centre and ever so politely ask to have your passport back, they will ever so politely answer that they cannot do that.

(Both quotes are from the terms of service that I found at the next step after this one. I don’t think a direct link is possible, but I invite anybody to try.)

So all things considered your options boil down to either hope it is fast enough or submit the application after coming back from Rome. Considering they say three working days for the express application, chances are probably high that it works. Compare, for example with Russian visa that typically take weeks to months to process.