I'm flying from Cork to London with Aer Lingus and then London to Los Angeles with United after an hour layover in London. United Airlines isn't letting me check in to my flight, saying that my "first flight is with Aer Lingus, check in with them." So how do I get my boarding pass for my second flight with United?
You'll often get all your boarding passes when you check in to the first flight, so it's likely that Aer Lingus will simply hand it to you.
If the stars aren't aligned and that doesn't happen, just follow the purple flight connections signs when you get to London. You'll reach a transfer desk where you can pick up your boarding pass on your way to your connecting flight.
The Heathrow website has a flight connections planner that will give you a step-by-step guide to your connection.