This is a really tricky one, with several possible solutions - all with pros and cons.
I originally thought to show the borough as the "city name" - and initially started using "Kingston" for all street names, and not using the neighbourhood names like "New Malden" or "Tolworth". However, I started to think that most locals are going to think they live in "Tolworth" and use that when looking up street names rather than the borough name, which most of us don't really use that much.
However in some neighbourhoods, you can be even more granular. For example, you could say you lived in Islington - or you could break it up even further and say you lived at the Angel, or in "Stoke Newington" - both of which are in Islington. Again, when I used to live in Stoke Newington - I rarely said I lived in Islington.
Ideally I think we should try to follow the principle of letting locals define their city as they themselves call it, but people will have different ideas of what their neighbourhood should be called. This is especially true of the neighbourhoods where people identify their address more in terms of the closest tube station than the actual district name.
Taking the argument all the way to the extreme - if -any - part of London should be called "London" - you'd think it's the inner-core - and I doubt anyone's going to quibble about using "London" for roads in The City. But should we use "Soho" instead of "London"? "Mayfair"? "Westminster"?
As it's important to have some agreement/consensus I'll suggest the following possibilities, but I think we should wait to hear from more Londoners before setting one in concrete...
1.) We could use postcodes as the delineator - all N, NW, SW, SE and E postcodes are London. Downside: lots of neighbourhoods that you'd think would have their own identiers won't.
2.) We could use some sort of geographic delinerator - everything inside the inner ring is "London", or going even bigger, "ALL London boroughs are "London")
This still leaves open the question: how do we decide what to call the areas that AREN'T "London" that I alluded to at the start of this post...
You'll notice that I generally have been leaving the city field blank in hopes that a solution would become apparently later on, and I'd go back later and change it. A clear solution hasn't yet occurred to me though.