Born in Toxteth Park region of Liverpool...left there in 66 I believe

Family from South Wales....spent many holidays on the Gower, Rhossili

Ah plasticine.... Sadly we only know it as play-dough!

What era where you in Liverpool Spitz?  Did you live there?  I was living in North Wales in 1963-66 when I was tiny!  At that time, one of my Great Uncles was working for Ford in Liverpool, and they used to hire out vans and drivers to folks such as young musicians so they could move their gear to various gigs! Some as far away as Germany.... By 1964, those fellows weren't renting vans any more!  By 1966, I had my first Mini at three years old.  50 years later, I still have a 1966 Mini, much more glamorous than the old 850 Mini Van in olive green!

.... joke forming about getting your nuts in a twist..
....must..... resist.........

Cool….my family is from Wales and I was born in Liverpool…..
As for the nuts….I don't know..ram some plasticine in the wheel nut depression and show what you have

Wow, this like pulling teeth, as one of my favorite talk-show hosts used to say!!  You should have read on a bit further!  I showed the CN4s as well, very clearly showing the ball end.  Even an average Joe could tell between those two by looking at the wheel.  Surely you and I are at least average.

By the way my family is a mix of Liverpool and North Wales and so's my English! That and the top private school in Bagdad, Iraq before all the Ba'athists crawled out of the septic tanks! That school had me so far ahead of American schools in... of all things, English!

Ahh the joys of the written word...

I've re-read your post a couple of times Jemal, and I've still drawn the same conclusion (maybe my English English is different to your version) but after your explanation I understand what you're on about.
There are three basic METHODS of holding a wheel on (round, taper and sleeve) but many STYLES of nut, which can vary enormously as discussed - even between different sizes of wheel from the same manufacturer. 
Had the OP gone out and bought a set of CN5s and graunched them up tight without showing us a photo of the seats, who would be at fault if they were the wrong ones - the OP or the tech guy ?

Good point AL. My Mini has alloy road wheels and a steel spare. I've never tried on the spare... something else to do when I'm physically up to it.

"Or tries to use a closed end nut on a steel wheel with a hub cap"

How true, back in 1997 @ Sieverville TN, l was getting ready to go on a Rallye at MM East(including a drive. By of Dolly Partons birth place and sure enough) l had a flat tire, no problem had a steel spare, so took off the slot alloy and when l tried to tighten up the stud, no luck, the closed end or covered end of the lug nut would not let me draw. It up, Bummer, missed the tour but later was able to find 4 open end lug nuts from the vendors present

So any Newbie should carry 4 open end lug nuts, in fact wire them to the steel spare in your boot

Big AL

I maintain there are 3 STYLES Alex.... of all people to miss-understand my queen's English, I can't for the life of me figure out why you seem to have so much trouble!  Again, the 3 STYLES are the radius or "ball" seat, the 60 degree V-taper seat, and the "sleeve" style you mentioned.... yes those ARE wheel specific... and I ask people to measure the shank diameter... It could be anywhere from 9/16 (.5625) up to 11/16 (.6875), with the most common being 5/8 (.625) ALL INCHES!  Now, to be very clear, those dimensions are the holes in the wheels, so the lug nuts will have an OD of about .555,  .620, or .680, again all in inches.

This post had nothing to do with the "sleeve" style, but since you brought it up, I've explained it.  And of course if one miss-matches the stud length with the requirements of the wheel or tries to use a closed end nut on a steel wheel with a hubcap, there WILL be issues.  So much of this involves common sense, which seems to be really lacking in the modern world!  Shall we talk about the thickness of the wheel flange itself.... the length of the sleeve that a particular wheel might require?  With unusual old wheels, sometimes the only option is to buy what's available and cut down the length!

And make sure they don't over tighten. 44 foot lbs. 

Mini Mania just announced a sale on wheels, lug nuts and centre caps

The ad comes with a lot of pics especially lug nuts and their specs

Check it out

Big AL

I once started to count up the different types of Mini wheel nut, and gave up after a dozen.

Nuts are very much wheel specific, though there are many that share the same design.

We'd need to see photographs of an undamaged wheel before making an educated guess.  Even different sizes of the same brand can use different nuts...

My 5" x 10" MWS Minilights use pre-84 round seat nuts (and need the large base version of them) but the 5" x 13" MWS/JBW Minilights use a large seat 60° taper nut.  Use of a small seat nut on either wheel leads to damage.

Just from the factory Rover Cooper Minilites use a unique nut that is very fragile, Mini Special (Exacton) use a unique nut, Sportspacks have their own fragile nut, 10" and 12" steel wheels with bolt-on hubcaps use their own design of nut, but though they look the same to the the untrained eye they are not interchangeable as one is round bottomed and the other is 60°.

Once you get into sleeve nuts for aftermarket stuff, it's a minefield. 

For example; the nuts for a drum brake offset Minilite are different to those for an S offset wheel.  Use the wrong one and the wheel will either not tighten at all as the nut bottoms out on the drive flange, or will be sort enough not to support the wheel properly...

For a Tech Support fella to blithely state that there are only three nuts is borderline dangerous.

There's no mystery here... All have 3/8-24 thread, and there are 3 types of "seat" styles... where they clamp and center the wheel.  My 13" Minators use CN5 nuts which are common to more modern wheels, just a 60 degree V-taper.  I could see someone could have put the round "radius" seat style and they would "work" as long as tight, but would eventually damage the wheels, specially if over-tightened.

Look at these CN4 nuts.... See the difference?  Now look at your wheels!

It seems to me that the shop left the wheel nuts loose if you have been driving the car that long with the same wheel nuts. A picture of your wheels and current nuts would help to determine what style you need. AFAIK there are four basic styles (maybe more) the commonly used tapered seat ones, radius seat ones, sleeved ones and the stud ones. Look on midland wheels site (now johnbrownwheels.com) as cc suggested as there is a single page devoted to wheel nuts and determine what you have and what you may or may not need.

I had 10" Minators and used these nuts

//www.minimania.com/part/CN4/Wheel-Lug-Nut-Crome-Acorn-Wide-Seat-Stock-Radiused-Tap

I would think if there's a noise, the wheels must be loose and are probably now ruined. You could still have the wrong nuts. There are many kinds of wheel nuts. The Midland Wheels has a good selection.

So, my Canadian-spec 1972 Mark 3 went into the shop with some brake issues. Brakes now fixed, but they say, after driving it around, they noticed some wheel noise, and have determined I have the wrong lug nuts. Now, I've been driving with these lug nuts for (I think) at least five years without incident. On the other hand, this doesn't sound like something I should take a chance with. Unfortunately, they appear to be having trouble locating the right lug nuts, which rather mystifies me. I'm running on 12" Minators. Any suggestions/ideas/comments? Thanks in advance...