https://pharmaphorum.com/news/russia...covid-19-shot/
This article should answer both questions whether the US FDA or EMA for the EU, at least for the Sputnik V vaccine. I doubt the Chinese would bother with getting US FDA approval for their vaccines considering widespread ant-Asian sentiment expressed or trust issues in the general public that wouldn't line up to take up such a vaccines and an Biden Administration expressing misgivings on certain issues with China.
At the US FDA can look into companies they are familiar with and have facilities inside the United States they can inspect. It's easier to go after companies like AZ, Moderna, Pfizer, Novavax and J&J if something with vaccines goes horribly wrong, you can't do that with the ones made in Russia or China.
The EMA may entertain using vaccines from Russia and China, but the US FDA is very risk adverse.
Other countries have their own standards so they are free to decide what data sets work for them, if they trust data from the US FDA and the EMA, then good for them. No country should rely on standards and decisions made from the US FDA or EMA to make the ultimate decisions, let alone have an increased confidence in a particular vaccine regarding the health and safety of their citizens.
The US FDA, for an example doesn't care for what the EMA or WHO does, they want datasets from participants from inside the United States, so Oxford-AZ will need to present their own data from US trials for consideration. Even J&J/Janssen and Pfizer-BioNTech had to present data from US trials even through they had assistance with their European partners from regulated countries.
It will be a difficult hurdle (or impossible) to get trial participants inside the United States to test out Russian and Chinese vaccinations.
If some HK residents will be getting Pfizer then its assumed it was approved and acquired legitimately.
The approval of what vaccines are approved for use and get distributed is mainly for the benefit for their own citizens and not so much how or where other citizens get their vaccine. They can easily mandate that foreign visitors use a mask in public spaces or anywhere outside their hotel or place of lodging.