Don't look for tour companies based in the UK as they subcontract to a Chinese company in any case.
Look for Chinese companies that can tailor a private tour for you like China Highlights or China Odyssey.
An alternative is to book your own hotels and internal transportation and hire private guides here and there. This is more economical and more flexible.
If you are unsure about your itinerary, just look for what organized tours and copy their program (excluding the Yangtze cruise which is a waste of time). Booking hotels and trains yourself is very easy. Using public transportation once there is very easy with subways that are fully bilingual.
One major downside of tours in China as the shoppings stops often disguised as factory visit or shops near to some attractions. Besides being a waste of time, prices at these shops can be as much as 10 times street prices.