Escorted tours
 

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.

Gate1 Travel is a US based company that offers an “Air and Land” option as well as a “Land Only” option for their tours. They have several tours to China (https://www.gate1travel.com/asia-pacific/china-japan).

I used them for my 1st trip to China, which was back in 2010. But since then I have travelled in China on my own, including to places like Lijiang in Yunan, which is a bit off the beaten track for western tourist. I never had an issue travelling on my own in China, except when I was leaving China at the Xian airport. The Chinese immigration officer skeptical that I was travelling by myself.

I have also used Gate1 for tours in other places, like Morocco and the former Yugoslav republics, and always been very happy with their service. I also know several of my friends who have used Gate1 and they too are very happy.

I too suggest considering organising everything yourself. With Trip.com, Booking.com and Hotels.com all working for China, it's actually surprisingly easy. Trains and metros are all straightforward (provided you've done your research), and if you use Didi then urban road transport is also simple.

​​​​My one caveat would be if you're going anywhere properly rural, but if you're open-minded and not in a rush then that too should probably be ok.

As mentioned above, the benefit of doing your own thing is avoiding the mandatory shopping stops, choosing your own pace, and it could well end up cheaper too.

Some high end tours are lead by famous professors, which arguably makes them useful, but unless you're a history buff, I don't see value in paying for the cost of the professor's time and accommodation expenses, especially in light of the fact that many of us --also well credentialed-- are willing to help for free.

More thoughts:

1. Beijing and Shanghai are both super easy
-book hotels on your own, and feel free to ask us for location advice
-we can also help you book transportation between the two
-the audio tour of the Forbidden City is really great, but if you're not into audio tours, simply seeing it (without audio) isn't so bad

2. If you want to add cities like Hangzhou to the mix, simply inquire, and we will provide our best resources
-e.g. my mom happens to be a strong enthusiast of the West Lake, so I made it a focal point in her China visit; we dropped $700 on a lakeside hotel, and a private boat; this was a much better use of funds than spending $2,000 on a Harvard professor to simply provide common knowledge

3. Join our Wechat group prior to your trip
-while I'm cautious about data privacy issues --and, therefore, refrain from discussing controversial topics-- we essentially provide free 24/7 concierge service throughout the region
-the concierge thing was never our intent, but when you have 130 members spread about the world, that's what you get

Ctrip is the largest OTA in China. Hope it could be of help.