Let's just say I was experimenting, and now I have a pan full of diced carrots and potatoes that have been boiled and way, way over-brined. They taste like... salt. It's very bad.
Basically dilution is the key, though a rinse and a short soak in fresh water will help a bit.
You could freeze them in small quantities and use them up in vegetable soup a little at a time. Don't forget that commercial stock has quite a lot of salt in, so reduce that accordingly.
Some thick stews could tolerate some mashed veg in them, again in fairly small quantities.
By mixing with completely unsalted veg you could make a form of bubble and squeak, but you'd have to mash/mix it more than normal to avoid having locally salty bits.
Osmosis is your friend here.
If you have not mashed them yet, drain them, cover with fresh water, wait, repeat - until the salt level is acceptable. I think that would be difficult with a mash since you'd lose vegetable with each draining. You will probably need to do this in the refrigerator to allow sufficient time for the salt to diffuse out without having spoilage. If you elevate the veg above the bottom of the container, the process works better, as the heavier brine tends to collect in the bottom.
I happen to do this with salty olives (which conveniently float, so no rack is needed to elevate them) in order to reduce the salt level to one I find pleasant. I'd suggest at least 12 hours per change of water, and then another 12 without water to let the salt left equalize unless you want a "salty center" effect.
If you are making lemon Greek potatoes like I was, you can rinse them, chop them in half, then soak them in water for 3-7 minutes.
There was enough flavour left after, and it seemed to get the heavy salt out of the outer half inch of potato.
Ultimately it made the whole potato taste much better, and the salt went from unbearable to unnoticeable with a dab of tzatziki.