You can use any of your US/Aus/EU passports at the e-gates. The UK does not have exit controls. There is no reason to try to keep track of which passport you use.

This may change in the future as the government intends to introduce Electronic Travel Authorisation, which would presumably be keyed to your passport. However there is no planned date for this.

I don't really understand your question about how the UK Government will see you. They really only care about your status at the border unless you want to work or settle. You are not required to carry a passport (or any identification) when you are in the country.