Do I need visa for visiting china from 29/06/19 till 2/07/19

Nobody can give you a proper answer as we don't know in which chinese city you'll stay, from which country you'll arrive in China and on which passport you'll travel.

We are British citizens. Travelling to Beijing and see great wall, forbidden temple, etc

This is perfectly fine, but keep in mind, as already mentioned by Thomas, that you have to arrive to Beijing from one and depart to the third country! Simply speaking, you have to arrive to Beijin from, let's say UK, stay in Beijing and after that you have to fly to the third country or territory like Japan, South Korea, Hong Kong and similar. Please, keep this in mind!

Thank you. I am travelling round the world and that's fine. How do I apply for the exemption visa?

Thank you again.
Bhav

The procedure is fairly easy. At the check-in counter of the airport from where you will depart to Beijing you will declare that you fly to China using 144 hours visa free transit. You will usually need printed ticket confirmation about your onward flight from China to prove your statement. Upon your arrival in Beijing you will head to a 144 visa free transit counter where you will apply for this convenience. You will fill a form where you will insert a long number of your onward flight and present the printouts of your flight and accommodation to the authorities. After that you will get a stamp in your passport and you will leave the airport to explore the city and the surroundings.