The winder does nothing else than pushing on the trigger button, then turning the rewind crank. Therefore the Mirror lock-up, if activated, should not cause a problem, as pushing the trigger button is the next step you do to take a picture. Bottom line, go ahead and try it, you cannot break anything.
I just tried it with my body. My first concern was that the winder would be in the way of the MLU control. It turns out it's not: there is sufficient room to insert your finger. Then, you can take the picture using the winder's release button or (better) the remote. Works like a charm.
As a side note, you do not need a film back to run this test. Just remove it if you have film in it.
Hope this helps