All you'll need to swap are the springs to lower the front suspension. There are a few things to check "while you're in there" like ball joints, bushings, bearings, etc. but none of that really needs to be replaced if the parts are still in good condition.
Lowering springs can be purchased from pretty much any one of the vendors and all lower the front to a height somewhere closer to the original designed ride height you see on preproduction development cars. If you run a search I think there are a few more threads comparing the different options and approximate drop for each brand of spring. I purchased my springs from DMCMW and they are made by Eibach... I'm very happy with the stance overall.
In this photo you can see my car (closest to the camera) compared to my friends car with a stock suspension. I am running the same Cooper Cobras as you in stock sizes. My friends car also has stock size tires but a mismatch of brands.
Note, I've also lowered my car in the rear slightly with the adjustable spring collars on the SPAX shocks I have installed.
Here are a few more for comparison:
Here's one more showing my car prior to lowering...for whatever reason I thought it was especially high even compared to some other stock cars. From what I've read there were a couple of different height stock springs fit to match based on variations in the fiberglass bodies.