The hypothetical became reality: my LX flight arrived 5 minutes before the UA departure...

I had to do security before getting to connections. I first tried the UA desk, they sent me to LX to rebook, joining a group of already 2 people who also missed the same flight.

At that point, according to the app, there was a UA connection available (admittedly with the first segment in first), that would arrive 5 hours late, same-day (10pm arrival).

The agent at the desk was extremely slow, eventually the first person there was given a connection via Lax (midnight arrival). This ended up taking 30 minutes, with the agent working over the phone to get rebookings done. I tried to call the premier line, but they claimed no more connections available - I ended the call, rechecked the app, and noticed there was still space available. At that point I got rebooked onto the LAX connection (without my approval), I asked for rerouting on a direct flight the next day.

That agent then sent me to the baggage claim to get the new itinerary. Apparently that was the wrong place, but I'm still waiting here for my luggage (I need some fresh clothing and my toothbrush, and don't have a rush to get into town).

Now, the delay was posted before boarding my first flight. The GA wouldn't switch me to a direct (more than 1 hour ahead of the amended departure time). Eventually at the transfer desk they were happy to switch me, but they weren't given approval because of offloading of bags (this was 25 minutes before new departure).

TL;DR: this gets messy. And LX and LHR staff seem to be powerless and/or incompetent.