Here is what you are legally allowed to do.
Trust me, once you have passed your test you will not want to stick to a 125 with all that up and down the gearbox every time you want to make progress. Something with a bit more torque and flexibility will make your commute much more enjoyable.
Most modern 4 cylinder 600s will have upwards of 70BHP, enough to get you to the legal speed limit in just one gear change and clearing 100mph with just one more, but low down (below 6k rpm) the torque is minimal and not really that suited to urban commuting.

Depending on what you want (all out sports styling or something a bit more street style) my suggestion would be to look at a 650 twin. Kawasaki ER6 has models with both styles and the parallel twin is peach of a motor with good torque lower down the rev range and is light and chuckable. Just the job for commuting.