Flew to Scotland last year, but into Glasgow, so I don't know how applicable my experience is for you, but...
Had a very smooth and relatively brief passage from disembarking the plane to passing through customs and on to the shuttle to the Glasgow train station. It was coming back that took more time--lots of tourists going home to the U.S., and several not paying attention or using common sense and thus slowing down the queue to departure from Glasgow.
Another possibility, though I don't know your schedule or budget, but have you considered flying in to London, then taking the sleeper train car overnight into Scotland? One would arrive in Scotland much more refreshed and less jet-lagged.