You can get work as a Teaching Assistant in London extremly easily. You can do this two ways:

1. Pick up your pbone and call a school that is local to you and ask them if they have any work for you (subject to police checks on yourself clearing)
2. Contact any Education Recruitment Agency and explain your situation. They will get you signed up and ready to go but it will be them that dictates what schools you work at and on what days. They would be the ones responsible for paying you too, not the school.

Teaching assistants positions in the UK are always available at both private and public schools.