Originally Posted by
corporate-wage-slave
I am in a similar situation, UK+France+NZ nationalities. There is no requirement to enter the UK on your UK passport, so in many cases I return to the UK with my French API in place. In theory you then move from right of entry as a UK citizen to discretionary entry, but the e-gates don't seem to do nuance. In any case I often use French API entering the UK but still present my UK passport. So long as you aren't claiming benefits other than the State Pension this is fine, see The Guardian about the off-topic problems this is causing (and will in any case be reversed in a few days in my experience.
So in your case I would home in on the Poland API, and then it doesn't need too much overthinking, at least at the moment. When we move towards ETIAS it may change, but I doubt it due to the huge passenger flows involved, vastly greater than the USA ETA processing.
Isn't the check-in system going to check that you have an ETA if your API is for a non-UK passport? I haven't tried this, but did hit the "computer says no", when doing the trip in the reverse direction, i.e checking in at LHR for GVA with UK API. As I am a Swiss resident, the system insisted on seeing a Swiss resident's permit, which as a Swiss citizen I do not have. The agent had to work hard to overcome this. I wonder what will happen when ETIAS comes into operation.
I do have to add that the modus operandi I described in my previous post was following discussions with a neighbour of mine, who happens to work with the Swiss border force.