Width is going to be the problem. A 5.5 or 6" wheel, which you would need for 165 or 175 width tires, will need front fender/wing cutting. Even 175/50 tires are not so easy to find anymore, and I've not seen a 175/40 for sale (or a 165 13" tire.)

Oops.

In my initial message I said "modern 13 inch low profile (SEVENTY series) tires.  I meant to say "modern 13 inch low profile (40-50 series aspect ratio ) on low profile wheels".  Smartphone writing small text does that.   Sorry.

Does this change any of the questions if it wil fit or not?
I imagine the ride will be rough and choppy.

Onetim, I'm way down in muggy Alabsma, but thanks for the offer.   I need to.update my profile.  This has been a good thread because it has dispensed some false info I've read. With the odd.tire size and the limited options available, it helps to know what works and what doesn't.  Late '70's I slalomed my BMW 2002 and my track tires were racing slicks with an aspect  of 40 and run on wide Chevy Vega wheels.   The street ride was horrid but the track.handling was great
  But the look wild, at least to a kid in.his 20's
 

Minibill, I would be happy to let your rip around my 13" wheeled 62 Cooper, see the modifications required. I cant tell where your live, but if your near Detroit MI, let me know.  

Can't see why you would be interested in different "compounds"....  (and as already noted, 175/70s won't fit.... (unless you run a certain model of Moke

Last time I was in the market MM was extremely uncompetitive with tyre prices ..but that has changed significantly (but Tire Rack does free shipping).

The range offered by MM covers anything you would want.  Falkens are a cheap, durable, less grippy but entirely suitable for touring /general road use.  Yoko 32Rs are boy racer (or gentleman competitor territory and price and durability reflects this.  Yoko 008s are (IMH   a good compromise for "spirited" driving.....

Shipping from the UK is no big deal and there are even more options there.

12s and 13s are also factory fit but there is a certain collective wisdom that these are more bling than functional.... And either can be fitted without any real hassles (read no body mods) if you choose the right wheels...

Cheers, Ian

I think Malsal's suggestion is a great one.  
If you can find someone/know someone that has a classic MINI with 13" wheels.....try it out first !

It would suck to do some "body work" to the car and spend the money and tyres and such just to find that 10" was right all along.

10" aren't really hard to find actually.....are they?

I think the word Doug and Scott are looking for is "harsher".
The shallower the sidewall the less compliant the tire will be.

I'll echo what dklawson said, but go one step further. In my experience, tires make a huge difference in how you experience the suspension. I've only ever had 10s on my Mini, but on my MINI I've had 16s and 17s, run-flats and regular. Sidewall height and stiffness make a huge difference. After a bought of back problems several years back I had to swap out my 17 summer run-flats for my 16 regular snow tires just to avoid wincing at every minor imperfection in the road. I wouldn't describe it as "bouncier" though... To me, the lower profile or stiffer tires really make every bump feel like a serious thud that leaves you feeling like you took a punch to your kidneys.

You will not get a 70 series 13 inch tire to fit under a stock Mini. The closest 13 inch tire to the regular 145 or the 165 10 inch tires would be a 175 50 13 which was fitted to the later Sport pack Mini's.
That being said if you go to 13's you will also have to cut the front fenders to clear and may need to use the rack limiters to stop them rubbing on turns.
Also the Mini was designed with 10's and the 12's are a good compromise but when you get to the 13's the driving experience changes, if you want 13's i would suggest you drive one with 13's first before you spend a lot of money and possibly regret it.
You can buy 10's shipped quickly to the US from most Mini suppliers for around $400 if you look around.

Another quick observation is that with 12 or 13 wheels you can run 8.4 inch brakes, which really work well in stopping the car.

Good info.  One other passion is classic motorcycles-- a 1973 BMW and a 1970 Hodaka.    I've gotten used to digging for and juggling parts.

 I get mine from Tire Rack.

Others will post definitive answers but I will offer the following observations.

With a Mini, to a degree the tires behave as part of the suspension.  Low profile tires may make the already bumpy ride a bit bouncier.

The low profile rims are clearly available for other cars.  However, you may have trouble finding rims with the right offset and 4" bolt circle of the Mini.

Quickie tire question to help future planning on this Mini project.