I'm 13 months into BA's trainee scheme.
When you complete the programme, BA assign you to the fleet and to the base where they need you.
As for aircraft that could be 737, 757, A319/A320 and for bases: Heathrow, Gatwick, Birmingham or Manchester.
Most people on my course anticipate being started on A319/A320 at Heathrow (could quite easily change!).
You do get to express a preference of fleet and base, but it doesn't count for much - ultimately the flight crew flexibility that exists within BA comes from pilots in their first five years with the airline (not complaining! - it's an incredible opportunity and I'll be delighted with whatever I'm given).