Are you able to travel slightly further afield than Birmingham? Jaguar Land Rover has a very good HR grad scheme (have friends doing it) and that is in Coventry.
There are some very big firms in Birmingham which also offer HR schemes e.g. accountancy firms like Deloitte, but I'm not sure if these are ones which ask you to move around every 6 months or so.
Also, completely NOT HR but if you're interested in doing recruitment there are some very big recruitment firms in Birmingham as well, e.g Michael Page in Brindley Place.