When in the USA, am I considered a British citizen working in the USA
or an Australian citizen working in the USA? Does this depend on which
passport I use to get sponsored?
For what purpose? While I'd suggest you asking an immigration attorney and a US-licensed tax adviser (EA/CPA licensed in the State you'll be working in), it is my understanding that:
Your immigration status depends on the passport you've used to enter the country (i.e.: you can enter under Australian E3 visa).
Your tax status (treaties) depend on the country of your citizenship and tax residency (i.e.: if you're not tax resident in the US, you can claim both British and Australian tax treaty benefits).
Whilst working in the USA, how are my capital gains treated, in the
USA and also other countries, where I have investments?
Once you become a US tax resident, all your gains worldwide are taxed by the US. You'll need to work with accountants in all countries to avoid double taxation, and you may not be able to avoid it completely. US Foreign Tax credit is a useful tool, but limited, and will not provide full relief. Once you're US tax resident - tax treaties are no longer helpful for most provisions (including capital gains).
You become US tax resident if you stay in the US for more than half a year, or if you become permanent resident. Most people on work visas end up being US tax residents if not in their first year in the US - then in their second.