The magnification will be 1:1, unchanged with that lens at 1:1 on any body.
The only difference is that the cropped body will crop the image.
Say you photograph some 15 mm object at 1:1 with both bodies.
At 1:1, the object will be 15mm on any sensor. The lens does what it does. And that is the meaning of 1:1.
But the full frame will be 36x24mm with a full frame body,
and the frame will be around 24x16 mm with an APS crop body.
But within that frame, at 1:1, the 15mm object will be 15 mm on any sensor.
Samples of this at http://www.scantips.com/lights/cropfactor.html
Now we might imagine that the crop body enlarges the object more, because after all, this 15mm object is nearly full frame height in the cropped body, but only less than 2/3 full height of full frame body. But the cropped frame is simply a smaller image (it is cropped), and it must be enlarged more to view it at the same size as the uncropped frame. The cropped body compares to using smaller film in film cameras.
1:1 means actual life size on the sensor. Any sensor, of any size.