In short, Cobra's leak....
They leak at the windshield, they leak at the doors. The trunks leak around the seals. The top leak and the side curtains leak. You will have a foot wells full of water that Flipper could live in, which sloshes like a wave pool on acceleration and braking. In general, the windshield wipers are ornamentation.
Remember these were originally British cars.....

Rainx outside and anti-fog on the inside glass surfaces are a must.
A good project is to add foot well and under-seat drains to the car. Boat shops have some slick little assemblies for this addition.
Maintaining a good cruising speed will divert most rain over the windshield but water finds its way in regardless.
Will it hurt anything? Most of the cars do not have water tight electrical connections. I have seen the instruments fog up over time. The ignition box needs to be kept dry. The carpeting dries out pretty fast if the weather improves.... the cars under-hood detailing is wiped out. You will want to put any luggage or anything you want to stay dry in sealed plastic bags.
Then there is the hydro-planning on the wide tires..... and lack of traction on acceleration which are both quite dangerous.
My wiring diagrams will be useful too after prolonged driving in the rain. Relays don't like water.
On the plus side, you do get a lot of thumb's up from those people passing you on the road. Truck drivers.... well that is another story....
My son went on some car spotters website this morning and there was a photo of a guy in a red Cobra in the pouring rain probably on the way to London.
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