Back in the day I was a massive fan, and still am, of 90s Fatboy Slim, Chemical Brothers, Lo Fidelity Allstars, Midfield General, Plump DJs, Crystal Method etc.
However I haven't heard a decent big beat or breaks album in years - am I just missing some great stuff or are the genres dead?

Breaks are still around but they've involved into more hardcore forms with Happy Harcore, Breakcore, and all the variations on breaks. I imagine classic breaks are still around but I'm not so much a fan, so I can't help. I just thought I'd say that I consider the breaks dead.

Why is it called Happy Hardcore? For that matter could someone give a brief description of the 'core genre and it's various permutations?

It's called Happy Hardcore because of the sound of the music... it's fast-paced, will sample at really fast speeds (the samples are even generally sped up to the point of having a chipmunk sound), it's upbeat.
When it comes to various electronic sub(-sub-sub?) genres, wiki is a decent tool but not really that great. It usually well give you a basic understanding until you actually hear the music:
Hardcore
HappyHardcore.com is a great way to find out about what's going on with the music. It's essentially the genre's pulse.

Breaks aren't dead, they've just changed into something different.

Check out anything by Hybrid and Audio Damage by Future Funk Squad.

These are much cleaner sounding, atmospheric breaks and in Hybrid's case they're downright symphonic, so if you're looking for that big Chemical psychedelic sound then this is not for you. There really isn't anything exactly like the artists you mentioned any more, and that's sad.

I'll check out Future Funk Squad (Hybrid I know of) but christ, breaks and big beat do seem to have bit the dust.
I may shed a tear.
Even if that is the case, I'll not be indulging in any happy hardcore just yet tho...

Breaks is not dead and it hasen't even changed into anything, I have no clue what these people are even talking about. They're completely stearing you wrong except Twinkie's suggestion with Hybrid. Breaks has never been called Happy Hardcore wtf? Don't even fucking try to listen to Happy Hardcore... Happy Hardcore is nothing relatively even close to what he's looking for.. Overseer, Lunatic Calm are a bit old like the artists you listed but if you haven't checked them out before, you should. Drum and Bass is now throughly evolved and you should check out some artists within that genre if you haven't as many of them are influenced by many breakbeat artists like some of those you mentioned. Just keep looking up some new breaks labels and you'll find some good stuff

With Tipper, LUX, Lawgiverz, Jason Sparks, and everything on Botchit, breaks are doing just super.