As Dragonfly has said, DO NOT use Fairy liquid on your paint, it will dry it out way too much.
Start as you have said, Dodo Born to Be Mild will be perfect, a very good shampoo.
Then Clay with your Meg's Clay Kit, this will remove the previous layer of wax.
Lime Prime will be the perfect polish to add a little shine to your finish.
Again, it sounds like you have the right idea with the Butter Wet Wax, or you could just buy Colly or Dodo and put a few layers on.
Don't forget a few good quality microfibres and applicators.
Chris