In the end I never bothered with In N Out burger.
Was a long day of travelling having to get up early to get across London on the train, and then go through the general unpleasantness of Heathrow airport (I don't like the way all the seating in terminal 3 is in one central area with the shops around). No lounge access as was just on Virgin PE, so tried my Priority Pass at one of the lounges but they said it was packed to capacity ... looked it from peering past the desk, and looked very average anyway. The queue for the other Priority Pass-accepting lounge was out the door so didn't even bother. Settled for a Boots meal deal, a seat with a power socket and watched the world go buy for a couple of hours.
Virgin PE flight was excellent (miles ahead of Singapore Airlines on the way over). Great seat and good food and service. Only complaint - not the airline's fault in the slightest - was the person sat next to me on the aisle was one of those types who didn't get up for the whole flight and got annoyed at me wanting to get past a few times to use the bathroom or stretch my legs. And the pay wifi wasn't up to par performance-wise.
Timed it at 1hr from landing to clearing customs queue. By the time I had made it to the front there was only a tiny queue behind so I'm guessing how quickly you are getting through depends heavily on how many flights land at the same time. Queue moved consistently, it was just massive. I'd probably have found it a lot more miserable travelling with kids, but as a solo adult it wasn't too bad. I just couldn't help thinking how much faster the queue would be if they had a line for people with onwards boarding passes wanting to "transit", for lack of a better term.
Decided once through that I'd just clear TSA as quickly as possible (didn't take long, I just don't understand why manners appear to escape all of them) and head up to the Star Alliance lounge and chill in the rooftop bar - on the VA flight I got an email from a colleague saying I had to check that out. Had a nice shower (with no wait, unlike Star Alliance lounge in Singapore which seems to have enormous wait times whenever I've tried) and then spent a few hours out on the terrace catching some late afternoon sun, enjoying a glass or two of the finest Diet Coke being the boring teetotaller I am, and having a snooze.
All in all not a bad experience, and I'd consider going that route again. I've had so many people say they wouldn't do the LAX route for love nor money, and certainly when I last went outbound on the old Air NZ Auckland to Heathrow service in 2019 it was miserable clearing customs then trying to race through security to make the next flight with no time to relax, I swore I'd not do it again. But having that longer connection meant no stressing and just taking some time to people watch, chill in the lounge etc made it all rather pleasant.