I would say if מיד would be "right away", then - באופן מיידי would be more of "in an immediate manner/way/fashion". It's quite a tricky one to explain. I can tell you that if you ask "When does the work start?" (מתי מתחילה העבודה?) the answer would be "Straight away" while if you were to ask "How does the job start?" (איך מתחילה העבודה?") the answer would be "Instantaneously". so if I had to find an English equivalent of it - I would say באופן is equal to adding -ly endings to Adjectives, not necessarily just when it comes to time. So יפה would be Beautiful and באופן יפה would be Beautifully. I hope I make sense of this tricky subject and someone finds it helpful. I'm no grammar expert, just a native:)
To Blacknoirschwarz: Your answer about באופן helped me immensely/ באופן עצום .