lens - How does background blur (bokeh) relate to sensor size?

This is a somewhat theoretical question.

Suppose I first take a photo of a subject using a full frame sensor DSLR, with a given lens (say 50mm prime lens at f/3.5).

Now suppose I exchange the camera for an APS-C sensor DSLR (with a 1.6× crop factor). I keep the same lens (same focal length, same aperture) and I step back a few meters in order to maintain the field of view (at least keep the same magnification of the subject). I now take a second photo.

Clearly depth of field will have increased between the two photos. But what about background blur (for example, trees at infinity)? Will I have the same amount of background blur, or will that have changed?

I have read somewhere that background blur depends on the physical aperture size. In this case the physical aperture (physical focal length divided by f/stop) stays the same. But should this number be taken in relation to the sensor size? In which case with the smaller APS-C sensor, the physical aperture will be relatively bigger, which would mean more background blur. This would be rather counter-intuitive as we usually consider it to be more difficult to get background blur on an APS-C camera.

Please give the reasoning behind the answer. I would answer the question myself using this background blur calculator but I can't manage to get it to run on my computer.