As I understand it, all vehicles that require you to look through the windscreen must have a screenwasher. If you can look over the screen, fold it down, or open it up so you can see through the gap, then it's OK without.
I suppose the MoT exempt status means you're not likely to have anyone check, but it wouldn't be good, if you had an accident, say, to have your car not according with the construction and use laws.
The person who restored my Mk A Bond was very keen on it looking original, and made his own 'washer jet' from a miniature matt-black tube, coming through the lower screen rubber and bent to point at the glass. It's absolutely TINY, but passed the MoT each time, when it was required.