I am a UK national and wish to apply for China visa in Los Angeles. Can I do this through the Consulate there?

According to the latest regulations, the applicants in the third country or area must provide their resident permit for employment or studying or any other kind of legal status (valid visa) in the third country or area. If you have valid document that allows you to stay in US legally, I think you can apply there. You might contact the consulate to confirm this.