I'm probably similar to you, OP, in that I consider myself a confident driver, but wouldn't want to drive in central London.
I was very wary of driving on the right for the first time, but I have now done it several times and it was fine, so I'd say just bite the bullet and go for it.
The first time, we went in our own car. That had the advantage of familiarity with the controls, but obviously it is right-hand drive, so visibility isn't as good because you are in the outside of the road rather than the inside. It helps to have a front-seat passenger because they can see better!
I have also driven in Europe and the USA in left- hand-drive hire cars. The cars took a bit of getting used to, partly because of a lack of familiarity with the controls, as you would get anywhere in a hire car, but also because things aren't in the place you expect to find them
(the rear-view mirror being the best example). But that aside, it feels more natural because you are sitting in the right position within the car.
Again it helps to have a front-seat passenger who can help navigate (or I suppose a sat-nav) so that you can concentrate wholly on the driving until you get used to it.
My other advice is to time your journey so you don't have to start off in the dark! And don't bit off more than you can chew - so if you don't drive in London, don't try to drive in Paris!