Lets keep it simple:
I know that the 185/60 r13 and the 215/15 r13 are about 21.5" in height. I have been told that they will rub the attachment point of the trailing arm. Is this true?

I have seen it on a car, but don't know what mods were done. I am not against cutting sheet metal, but am concerned about height clearance. Please if anyone has done this let me know how.

 

Big tires are going to interfere with your wheel arch at the front lower. Without proper clearance it'll bend your fender under. They'll also rub on your back inner fendr well.

Rim offsets and width matters.

Those are big tires for a Mini. Are you sure you don't need a 165/175 45 or 50 series in a 13? I had Sumitomo 13's on my Clubman but don't remember the size right now. Pretty sure they were 175/45 series though

 

the sumi's are 175/50 r13. Those are 20" tall. I know I will have issues with the front and will cut accordingly. I am most worried about the rear trailing arm.

 

I have Yokohama A539 175/50-13s on my car, with Superlite 13x6 wheels. No problem with the trailing arms, but I have finned brake drums with the 1 inch built-in spacer. The rear track is slightly narrower than the front, so the spacers just bring the rear wheels out closer to where the front wheels are. You'll def need wheel arches.

 

The rear also fouls the shocks w/o spacers.

The front fouls the bodywork due to rim width and offset. IE, on a left turn, the right front tread will contact the bottom of the front of the wheel arch and fold your fender into a pretzel....