I didn't find many posts on this issue (and fix) so I thought I'd create one for future reference.

2016 Countryman S ALL4 , 32K miles, bought in December 2015. No service issues until now.

About month ago, I noticed the smell like oil burning, exhaust, or gas inside the cabin. At that time I couldn't determine what it was. It started coincidental to a sloppy oil change so my initial diagnosis was to wait and let the spilled oil burn off.
I don't drive this vehicle much, so this was a few weeks. It never went away and it became stronger and more distinguishable as fuel. There were no drips underneath and smell would only happen when the engine was running. You could smell it outside the vehicle. It was strongest from the left side of the engine compartment, and it would permeate the cabin via the cabin air filter path.

I took it to back to the MINI dealer. They found that there was a leaking fuel pressure sensor and the fuel rail, and seals (specifically this part: 2012-2016 Mini Fuel Rail 13-53-7-630-645 | Mini Parts Direct , and 2010-2016 Mini Fuel Injector O-Ring Kit 13-64-7-600-869 | Mini Parts Direct) . They had to order parts and it was a few days to complete the work after initial drop off. The problem with the smell appears to be fixed now.

I was lucky because its an expensive repair if done by the dealership, however my state honors the 7 year Emissions Warranty, and since this was part of the fuel system it was covered. I wasn't aware of this and thought my warranty had expired after 4 years which is why I dragged my feet a bit to take it in first place.