Ok, slightly conflicting answers, but it sounds like failed wishbone bushes are most likely. Specifically the front bushes, but could be both.
In severe cases where there is free play they allow the wishbone/control arm to flap. Like a tank slapper on a bike. Feels gyroscopic and the front of the car wobbles with the steering wheel.
A quick sharp jab on the brakes can set it off almost instantly, or a bump on one side of the road that only hits one wheel, are most reliable tests to set it off but it can occur at any time. ...and at any speed btw, but lower speeds seem more common.
Some tests to confirm, put a foot on top of the wheel and push it for and aft, if the wheel moves easily with more than say 5 mill of play, knackered bushes.
Or, jack it up at the front, both sides on stands, get under there with a bar and a torch and lever the wishbones in the gap between the wishbone end and the subframe.
There should be play, but need a great deal of force to move them. If they slop about, as sounds likely, buggered bushes.
A valid mot, does not mean the bushes are sound.
If by some odd chance the rear bush is ok via a visual check writhe wheel off then a cheaper and easier option could be to fit poly, available for the front only.
Hth.
