Yes, the hub is identical
Same 5x112 PCD and same 66.5 mm centre bore: wheels from Audi S4 bolt on and centre on Mini Cooper without rings. You still need to check the offset and tyres, detailed below.
≈: width estimated from the original tyre size.
Wheel by wheel: offset and tyres
Each original wheel from Audi S4 is compared with the one from Mini Cooper of the same diameter: how much closer the wheel gets to the shock absorber or how far it sticks out of the wheel arch, and whether the tyre keeps the same overall diameter.
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≈6Jx15 ET40 · 195/65R15 — Check on the car
Compared with the original ≈5.5Jx15 ET45 on Mini Cooper: shock absorber side +1.3 mm, wheel arch side +11.4 mm, tyre diameter +4.3 % : fit it with the original tyre size of Mini Cooper (175/65R15), not that of Audi S4.
Comparison for guidance only (estimated width). Check in detail
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≈7.5Jx18 ET40 · 245/40R18
Mini Cooper has no 18-inch original wheel in our data: check the approved fitments and caliper clearance.
Before fitting the wheels
An identical hub is not enough: offset, width and diameter change the position of the wheel, and larger calipers can touch a smaller wheel. Original wheels often use bolts or nuts with a specific seat: keep the fixings intended for the wheel, with the thread of Mini Cooper (-), tightened to the torque shown on its vehicle page. Finally, check that the size is among the approved fitments for your vehicle. This comparison is for guidance only: have the fitting approved by a professional.
And the other way round: wheels from Mini Cooper on Audi S4?