I'd recommend a singlespeed bike with a freewheel rather than a 'fixie'.
A singlespeed has no gears to worry about, but has conventional brakes and a freewheel - if you want to stop, pull the brake levers, and if you find yourself going downhill you can just rest your feet on the pedals and the bike will happily coast along
A fixie is a particular type of singlespeed that has no freewheel - you brake by backpedaling, and you can't coast - you must constantly keep the pedals in motion. Suffice to say, they take some getting used to...
Singlespeeds are perfect for commuting in London since there are hardly any hills of note
Tomo, you can still easily spend a lot of money on a good bike even if it has no gears, but for your budget you might be able to find a good second SS on ebay or bike forum