Hi Sarah
The security check is done in the form of a "questionnaire", if you will. You fill it in and return it to BA. It takes about 6 weeks to process but I'm sure you'll be allowed to start training before it's finished.......
That form will arrive with your letter of appointment.
As for getting in to The Rivers / Cranebank, I don't think that would be a problem either. The bus driver that took me to the offices didn't look at my letter and I am sure that the people on the front desk will be able to confirm that you're meant to be there!