I'm not from London, so perhaps forgive me for wading in on this, but I have to say, it amazes me that something that could have so good has instead been so badly bungled. You'd think that a city with a higher population than any of Denmark, Finland or New Zealand would be able to sustain its own TV channel.
Indeed, BFM Paris seems to have done well in the two or so years it's been on air. Amsterdam, as well, has had its own TV station,
AT5, for nearly thirty years. It is worth noting that it has been in and out of financial trouble a few times; in 2012, the channel was acquired by a consortium consisting of a newspaper, another regional TV station and public broadcaster AVRO. In addition, it is also partially reliant on subsidies from the city of Amsterdam, alongside advertising revenue.
It is worth noting that their news output is perhaps a bit lacking: one 15 minute bulletin a day, which is then repeated on the hour every hour; there used to be more, but lack of finances has put paid to this. They do make quite a bit of other stuff, though, including a local Crimewatch-type programme made in conjunction with the police, a programme about the city's history, and a fortnightly interview with the mayor. There is a large number of repeats (understandable given the channel's shoestring budget), but all of the channel's output is themed around and relevant to the people of Amsterdam. It is also quite an important breeding ground for talent, with a large number of well-known Dutch TV journalists having once worked for AT5.
I mention all of this because I believe that AT5 is a perfect example of how you can do things on a shoestring, yet still remain relevant to a local audience, without having to resort to bought-in films and series repeats. I think many of their programme ideas, especially the ones I mentioned above, are ones which London Live would do well to emulate.