which companies will train me up as a trainee mortgage adviser/broker for free in London?

Answer 1

The one you get a job with :-)

Apply to all who have vacancies for "trainee mortgage adviser/broker" ....

NB = these days Companies are getting wise to people who only join them for the training & then leave .. you may have to sign a contract to stay with them for 3 years (or more) - and agree to pay back some of the training costs if you leave 'early'.