Well the London area is in one sense of the most difficult areas to find instructing work, while the schools in the London vicinity may benefit from the financial affluence of the area, it doesn't seem to transpire into a demand for instructors.
Well the way I have always found instructors work was by driving around all the nearby schools (and not so near) and handing CVs in in person.
This method got me my first part time position and my most recent.
If you want to do an FI course, I wouldn't rely on being taken on by the School you trained at, it does happen but not always.
There did seem to be a bit of a demand for instructors in Scotland in recent months.