2004 Mini Cooper worn inner CV Joint housing?

Which axle are you replacing? Or are you planning to do both.

If you replaced CV axles before then this mini won't be to much different. It involves removing the wheel, brake caliper from the knuckle, disconnecting the suspension & steering, wheel speed sensor, axle nut, and removing the axle and reinstalling everything.
I would recommend using air tools to do this repair, especially for the steering, suspension and axle nut on the end of the axle.

Some fasteners use nylon integrated into the nut to prevent the nut from coming off of the steering & suspension parts. This nylon makes it very difficult to remove with just hand tools.

The job pays about 3.0 hrs to replace both axles or 1.5hrs for the left side and 1.8 hrs for the right side axle. This is with shop equipment.

For parts most shops will just replace the axle, so that is the only part you'll need. If you what to do the repair by the book then replace the axle seal with the axle. Doing this reduces the risk of transmission fluid leakage after replacing the axle.

FYI
Some parts that will cause a wobble with the vehicle in motion.
Unbalanced tires (front tires usually), bent rims, tire internal construction is separating causing the tire to become out of round and worn axle shafts can all cause wobble.