It will still pass its MOT if you remove the screen and wipers and pop on aero screens . No idea if it will pass without aero screens. Just in case you have a problem with a date for an MOT test before you get hold of a new screen .
Yes, it is (or was) possible to get a car MOT'd without a screen, Steve Simmonds did exaclty that with mine when he had it to make a new hood. He managed to break the screen somehow, and whilst waiting for a replacement (not heated, so not desperately expensive) the MOT ran out, so he got it MOT'd without a screen. The tester simply wrote "Windscreen not fitted" in the comments box.