I wrote a looong reply yesterday and it got lost. I'll put some general info in first and then put some links in later.
As you're going to Tokyo at least twice - firstly, I highly recommend getting Kodansha's Tokyo City Atlas - a Bilingual Guide (ISBN4-7700-2503-3) it's 2,100yen but it will make Tokyo into a much less scary place. I promise!
Second - yes! Go out on the town that night! But don't get too freaked out with the prices. OK, it all seems expensive as most of the good clubs will charge a 6,7 or even 8,000 yen admission charge...but...you do get 4 or 5 drinks tokens included with that. For cocktails that would normally cost you at least 1,000yen each. For your money you'll get great music, decent airconditioned surroundings (the gay clubs will throw in fab entertainment, even cabaret) and, this is the important bit, as most of the kids there are marooned until 5:30 - when the trains start - there'll be a comfortable chillout room too, so you can snooze if it all gets a bit too much.
If you are really into this idea, jib71 mentioned that he could give you more pointers, and I'm sure others here could too. The club scene is in a constant state of flux, so the places I've been to aren't necessarily there anymore (or any good if they are) - I took tips from this website the first time I went to Tokyo - I got friendly with a few people on my first night out and just went on their recommendations afterwards (you may need your passport for some places, so do take it with you). Ah yes, most clubs will have coin lockers, if you plan on visiting a sento you can stuff your towel in one.

There are Sentos (public baths) that are open all night where a lot of marooned older people end up. Some of them are kind of entertainment centres in their own right and you can certainly enjoy a few beers here.
(list of sentos and reviews here - scroll down to the bottom to see the 24 hour places, under the heading 'bed, bath & beyond')

Another hang out for people who can't afford a taxi home are the all night cinemas in places like Shinjuku. - it'll be the same film over and over and lots of snoring mingling in with the surround sound.

Every time I've been to an all night club, they've allowed me to leave and return later, which means I've had late night suppers at the Izakayas that are open through the wee hours.

Another option if you do need a lie down are the capsule hotels - more options for guys than girls with this one. If you read the description of the 'Green Plaza' in the 24hr Sento section it shows that there is a male capsule hotel inside here too.

And if you want a change of scene, then there are the markets at Tsukiji which will be in full swing at 4am.

It's all so very different to a night out in London! Same sort of price perhaps, but way less scuzzy.

You can go to NRT at 5:30 and, once checked in, you can rent a day room for a couple of hours - here you can take a kip in a bed and have a shower.
Here are maps of where they are in Terminal 1 and Terminal 2 info and prices here

The LH seats in Y have those little wings for your head, not so good food (don't have an onigiri rice ball on the way TO Tokyo - they're OK on the way back. Trust me on this one!) and a lousy movie selection. You want to tire yourself out for the trip to London. Your body will just assume you've submitted it to a decent night out and groan and whine accordingly - it won't feel so much like jet lag as the horrors of the morning after the night before.

Go on, go for it!!