For the "trend fixie" look, you also need:
much narrower bars, cut down at the inner end of where the grips are now.
a five-solid-spoked front wheel in white or some other bight colour.
a rear wheel with a very deep section rim and normal spokes, also in a bight colour but different from the front.
all black tyres or some other colour combination.
double toe straps and toeclips.
the saddle needs pointing down.
Also you need track ends, but that would be a job for a frame builder.
And anyway, these 'Noddy' bikes are not "London type", they're everywhere, worldwide, I don't they started out in any one place.
If anything the trend fixie look started in New York, but London's fashionista have taken it to a level beyond anywhere else. Burberry pads on the top tube?! Stupid twisted spoke patterns? Deep section rims painted lairy colours? Eight inch wide straight bars (usually horribly expensive Nitto ones cut down)? Only in London will you find them altogether on the same bike, ridden by some weirdly dressed pillock who claims his mount is a retro track bike.
The odd thing is that back when bicycle couriers back in London first seemed to get heavilly into fixed wheel as a trend the chosen mount was usually an MTB. Yes it was fully rigid and flat barred, but so were all MTBs back in the eighties. Then the trend for retro track bikes drifted across the Atlantic. Then at some point it seemed that a competition to build the stupidest bike began.
I really can't see the point in any bike where any particular fashion statement takes precedence over functionality.