First step is to get yourself on the waitlist for NHS services. That could take a while, so do it sooner rather than later.
Then yes, titration will take a bit of time and potentially a few expensive private sessions. The £400 should just be for the first session and then reviews will be approx £150 til you're stable. On top if that, you will have to pay private prescriptions costs. I'd say realistically, you are looking at about £1200 -£1500.
Once you're stable on meds, you will no longer need regular psychiatrist appointments. However, if you end up on stimulant medication, you can expect to keep paying for those privately til you get an NHS appointment: iro £100 pcm. However, if you end up with non-stimulant meds, your gp might be happy to take over prescribing.
Did you suspect you had ADHD before the dx? I'm only asking because if its all new, then there are so many gains you can make just by learning more about ADHD. Once I started finding out what ADHD was then a lot of my co morbid stuff - mainly depression- eased significantly: had lived with such crippling shame and helplessness for so long.
So maybe don't worry about meds and going private for the moment and just find out as much as poss about ADHD and see what changes you can make while waiting for NHS appointment?
If you do start thinking about going private, you need to be prepared to spend that big chunk at the start and then potentially ongoing prescription charges til you get seen by NHS.