You really don't want a PPL logbook if you are going to do a lot of hours in multiple aircraft classes. Having multiple logbooks is also a pain and you don't need the hassle.
Whilst you're on single figure hours, just spend the £20 or so on the best multi-column logbook you can find and copy over what's in your current. If you've not changed school, you can always ask your instructor or CFI to re-sign it, but that doesn't really matter because you can just hang onto the old PPL logbook for reference if they don't.
I had the sense to do this at about your age and with about 20 hours to copy up; I've been glad many times that I did - my only regret is not starting off a bit neater!
It doesn't really matter whether the one you buy is called "JAR-FCL compliant" or not, just make sure it's a big one that has columns to cover everything that you might be flying in the future and the recording requirements for any licences you might be applying for.
I doubt that any professional training organisation will give you a logbook - it's up to you to decide what you want, buy it, and look after it. Some may have a preferred logbook, but I'd not worry about that too much, they'll accept any professional logbook.
For what it's worth, I think that many people think that
this is the nicest professional logbook on the market at the moment (and will probably replace my current one when it finally fills up in another 500 hours or so) - although personally I'd be happy using the FAA professional, AFE professional, or the Pooleys non-JAR logbook (which is closest to what I'm currently using and very happy with.) That doesn't mean that there aren't plenty of other perfectly good logbooks - I'm just expressing an aesthetic preference.
Oh yes, and buy something that you can get a leather cover for - it looks better after about 10 years of knocking about your flight bag when it finally gets left on the shelf at home.
G