As others have stated, you need to find a way to enter the UK from outside the EU, so that your UK work visa will be activated.
To answer your first question, though, API doesn't create the problem of "slipping through" as you may fear. There is still no exit check in the US, only the airline. If you have to enter a passport number online in advance, you can use either passport--you still need to check in with your passport at the airport so it doesn't matter. If you are coming to the UK longer than you would be allowed as just a tourist, they probably will be looking for a UK visa, so show them your Canadian passport with that in it. Still, of course you will have your US passport with you also (since you have to have it to come back to the US), so you can always show the airline both.
I can tell you that swapping passports mid-flight is common and allowed. It's entering and exiting a country on the same passport that you have to be careful of, not exiting one country on one passport and entering the next country on a different one. From personal experience, I have frequently checked in with my US passport in the US and then arrived in the UK and entered with my Canadian. The UK needs to see that I have the right of abode in the UK (which is in my Canadian passport) ; they don't care that my US passport is in APIS. They can clearly see that I am the same person, name, and date of birth, on either passport.
