Well, yes. That's what a Colby is :-) (with a cold water wash, that is). I'm not sure it would make sense to do it with a warm water wash, though).
I think you need to think clearly about what your goals are and what each step does. When you wash the curds, what you are doing is reducing the lactose level in the whey outside of the curds. If you use warm water, then the whey outside the curds will be *less dense* than the whey inside the curds. Also, because the lactose level is lower, the whey will be even *less* dense. This will pull whey out of the curds. The overall effect is to remove lactose from the curds.
With less lactose, the culture has less food and so can't acidify as far. If you do it correctly (for a gouda, for example), the culture will "bottom out" at a pH of about 5.3 - 5.5. What that means is that it completely runs out of sugar (lactose) and can't produce any more lactic acid. No matter how much longer you leave it, it won't get any more acidic, because there is no lactose left. This is why some Gouda (and Parmesan, which is similarly fermented out to completion) is lactose free.
Just to be crystal clear, the purpose for a warm water wash is to pull lactose out of the curd so that the acidity of the curd bottoms out somewhere between 5.3 and 5.5.
Cheddaring has a completely different purpose. The point of cheddaring is to drain the curds of whey before you put them in the press. This makes it easier to dry out the cheese, which will allow it to age for a long time without spoiling. Because the curds are drained in big slabs, the compress under their own weight and texture the curd. After that you mill and salt the curds and put it in the mold. This creates a crumbly texture in the cheese.
One of the key points in cheddaring is you are salting the milled curds. The high salt content slows the culture down to nearly zero action (though, not actually zero, as I have found out). So you are at about a pH of 5.3 and you salt the curds. Within the time frame of the early aging of the cheese, this halts the production of lactic acid production and keeps your cheese from getting too acidic, even though there is a lot of lactose left.
From my point of view, these two techniques don't really mesh logically. With a gouda, you bottom out the fermentation at a high pH, which retains a lot of calcium. This gives you a sweet, pliable, smooth paste. With a cheddar, you aggressively drain the cheese and then halt the fermentation with salt at a medium pH. This loses calcium and gives you a dry, crumbly, textured paste. If you did both, you'd be working against both goals. Obviously *something* would happen, but I'm not convinced it would be that valuable. I think you would be better off doing a washed tome technique. I think you would get a very similar result with a *lot* less work.
But you will never know without trying it. Cheese is pretty hard to predict. There are a lot of very well thought out, reasonably ideas in cheese making that are utterly wrong. I've learned long ago not to assume that my reasoning is a good predictor of reality.
