I think I Seoul rivals NYC in terms of the 24/7 nightlife or entertainment that isn't Las Vegas. When I was there you could literally go out every day of the week if you wanted to whereas NYC does settle down during the week.
Seoul is a bit sanitized, though. I don't think you can replicate NYC's "busyness" or grittiness which is why I love it here.
At least for London, this is simply not true. Sure, you may find there are some things open late, like maybe a gas station off The Highway, or casinos devoid of anything interesting beyond buckets to sink your money into.
And that's what it comes down to I think. While London (I can't speak for Berlin and Tel-Aviv, I've never been) may be a big city, it's far from interesting at 3am. Hell, in most areas it stops being interesting around midnight or 1am, when most pubs close. You'll have a hard time finding anything that's more interesting than a seedy chicken shop at that point.
That the only public transport available 24/7 are buses, most of departs perhaps once an hour, probably doesn't help making a city active 24/7.
On the other hand, in London you'll see bankers having a large pint at 8am. So I guess there's that.
When I used to frequent London (which would be 15-10 years ago), there was stuff to do at 3am, clubs, etc. It wasn't everywhere, but you didn't have to look hard. The tube stops running at some point, but the buses take over and are actually useful.
When I lived in NYC, it was the basically the same. Most subways switch to 1/hr after midnight, most parts of the city are sleeping. Union Square wasn't interesting at 3am, neither was Madison Square Park or anywhere around Central Park. There's stuff in SoHo/Tribeca, but not that much. At least that was my experience.
No NYC subway lines drop to 1/hr late night. Some lines stop running entirely, and others switch to making local stops, but the scheduled late-night headway is 20 minutes between trains [1] on all lines that run at night. It's been that way the whole time I've lived here (a decade) at least, probably longer.
Obviously this can be affected by track maintenance, police activity, etc but that's not the norm.
Other rapid transit systems in the area are a bit worse, but still not 1/hr. PATH late-night headway is 35 minutes; Staten Island Railroad's is 30 minutes.
The commuter rail systems (LIRR, MetroNorth, NJ Transit) may indeed drop to 1/hr or worse per line, but that makes more sense for heavy rail.
Berlin's U/S-Bahns certainly give the impression of being open 24/7 -- I've never had problems getting to/from clubs or airports at any time of day.
London can also claim a useful 24/7 service, with the night tube starting a couple months ago and the third and fourth lines (Jubilee and Northern) just opening up this week[1].
It's been a few years, but I remember getting stranded at the U-Bahn station a few times when I was in college because we got out of the bar too late and having to walk home or wait around for daylight.
