Assuming you have the original, standard CD Boost unit, you can get one of these, on eBay, for about £40:
Car Stereo USB SD AUX MP3 Player CD changer adapter-Mini Cooper R50 R52 R53 https://www.ebay.co.uk/itm/Car-Stere...3D222343737556
That is what I did as I wanted to play music on USB memory sticks. It plugs into the back of the head unit, where the 6 CD boot changer connection would be, and the head unit sees it as that. I had a 6 CD changer (I have Harman Kardon) and had to remove that.
If you want WiFi connection and hands-free phone capability there is a more expensive version that provides that, costs about £55:
Car Stereo Bluetooth handsfree A2DP MP3 CD changer adapter-Mini Cooper R50 52 53 https://www.ebay.co.uk/itm/Car-Stere...8AAOSwymxVPx15
There are numbers of these adaptors out there, for example GROM. But this was by far the cheapest, it works perfectly and is quite straightforward to fit.
With respect to auto stereo shops, I called numbers of them and found they were either totally clueless or wanted stupid money or wanted to sell me an aftermarket head unit, which I didn't want.
NewMINIStuff have a guy Mikey who is a guru on MINI sound systems and they sell an equivalent unit. NewMINIStuff worldwide mail order accessories for the MINI
Pic of mine installed. As you can see I've swapped the standard tiny cubby arrangement for a proper shelf.