I have had the same problem, that my consulate does not provide courier or agent service to arrange authentications from my home country.
I'm not from UK, but the basic process is same for all.
The original document needs to be authenticated in foreign ministry of UK (I assume in London). They can prove, that whoever signed the document was authorized to do so.
There is a catch, that the original document may not necessarily be signed by the British consulate in Chongqing - they may only act as an agent to fetch it from some bureau in UK. Have you confirmed this? Between the lines I read that you don't yet have it.
If the original document was signed by some other entity than UK consulate or other bureau directly under UK foreign ministry, it may be need to be notarized first in whatever city it was written, and only then authenticated by foreign
At least in my home country, a consulate is not qualified to tell whether a person is marriageable or not - they can only authenticate some other bureau's signature.
Next step is to authenticate it at Chinese consulate in UK (again I assume London). They can prove, that whoever authenticated it at the foreign ministry of UK, was authorized to do so.
There is no entity in China who can authenticate the signature from UK's foreign minstry - the Chinese consulate in UK is the only place who can do that. Simlarly Chinese bureaus in China will not accept anything that is not authenticated by China's consulate in UK.
Then you are done - save Chinese translation, which in my case was done internally by the bureau
that registered our marriage in China - specifically Baoshan, Yunnan.
You can opt to mail the certificate (with needed support documents - not sure if your passport is needed) to UK and have someone else do the authentications there on your behalf, and then mail it back to you.