a 50cc can be plenty nippy enough, a 2 stroke model once derestricted will generally travel at 40-45. a 4 stroke model once de restricted will pull to around 40 just slower.
18 miles in london will see you rarely get past 40 so 50cc is fine, commuting wise, once your happy to lane filter you will find it a blast and quiet possibly enjoy yourself, any ped will filter like a dream on litle wheels.
If you have a full car license and have held it longer then 2001 you can ride a restricted 50 without a CBT. And lets face it every 50 out their is derestricted so dont sweat that haha.
Now for costs and figures, buy a good brand bike, piagio, vespa, gilera, yamaha. Reason being is resale will stay higher longer. you can get a sym for a few hundred but it will be worthless when you sell it. Be safe and use a big chain.
Bike gear wise. Waterproofs are all thats needed, leathers will do nothing at london speeds other then be costly and uncomfy, a good jacket can be had for