Good choice coming over the bike side, plenty more fun than a boring car - even a little 125cc bike.
As you know, the CBT limits you to a max of 125cc, for 2 years, so I assume you are looking for something like that and are not intending to do a full bike licence. (Got mine out the way before going to Uni and becoming riddled with debts " />)
Commuting and central London...a 4 stroke motorcycle is the way to go then.
Avoid 2/4 stroke sports motorcycles, lower position and a revvy engine (2 stroke) won't be much good going through rush hour traffic and your view would be limited of the traffic in front. 2 stroke motorcycles (RS125, GPR125 etc) require higher maintainence then a 4 stroke, piston rebuilds and engine problems wouldnt be good - especially on a university budget .
Avoid Chinese make motorcycles, there are some 'good' ones out there but this seems very hit and miss, and in my experience (a AJS 125 while my bike was serviced) most are a sack of shite.
That said, some bikes you'd possibly like to look into:
I found the CBF to be my favourite (I've rode all 3, and owned a CBF for a year) It's perfect for taller riders, but a smaller person wouldnt struggle too much. The high view and relaxed position give good view of the road ahead. You also get Honda build quality and peace of mind with it, a good condition one on a '58 reg can be found for £1200 - £1800 2nd hand. However, the YBR and CG are both capable and proven bikes, and you cant go wrong with them - I just found the CBF a little more stylish and a taller ride...engine difference - nothing.
Edit: I missed one bit, insurance will be dirt cheap on any of the 3 125's I've listed, and most others. 4 stroke 125's can average from 60 - 120 mpg so you shouldnt have any problems with that. I would easily get over 200 miles a tank from my CBF125, a fiver would easily last the weeks commuting to college for me!
There's plenty more I've probably missed, I'm a bit new to this site. But if you have any other questions don't hesitate to ask. Apologies for the essay " />
ps: Photo whoring of my CBF is attached, miss it