As another commenter as noted, J&J is the company, their vaccine is called Janssen which is covered (but maybe not administered in the UK) [0]
Oxford/AstraZeneca is licensed/sold under several other brand names [1] including Covishield and Vaxzevria but so far the brands are used interchangeably (as long as the batch wasn't bad.. this makes sense, no?) Given this is British, the other brand names aren't particularly relevant to their population (they got the original). Covax, as far as I can tell, is a programme to deliver Oxford/AstraZeneca vaccines (no matter the brand).. perhaps you meant Covaxin (an India developed vaccine)?
Vaccine mixtures haven't been explored much (this would mean restarting the trails again mixing and matching, which is tricky given people are either completely against, or want the vaccines yesterday).. but the WHO has cautioned against [2]. Never the less several countries are mixing.. we'll see the results as they play out. Oxford(!) did find that mixing AstraZeneca + Pfizer was more effective than double AstraZeneca [3]
[0]: https://www.cdc.gov/coronavirus/2019-ncov/vaccines/different... [1]: https://en.wikipedia.org/wiki/Oxford%E2%80%93AstraZeneca_COV... [2]: https://www.msn.com/en-us/health/medical/who-warns-against-m... [3]: https://www.cbsnews.com/news/study-shows-mixing-astrazeneca-...