I have started checking a Mk2 Cooper S mechanicals.

I have the above with 59/19 tooth differential.

Will I need a special speedo pinion and are there any benefits or should I change it to original 62/18?

OZ built minis with either 3 or 4 sp remote change boxes used the same speedo drive ratios at the gearbox end and changed the speedo head internal odometer gears and speedo needle bits to match the diff ratio and wheel/tyre size to the distance travelled and show 60MPH.
Later OZ cars with rod change boxes used a different speedo drive pinion assembly - 2A 3720 vs 2A 3620 for the earlier remote and pudding stirrer boxes.
Was the ratio different? I don't know. Parts books don't give any info on the ratios.
For OZ built MK1 S cars the speedo cable turns 1280 revs per mile which is also 1280 RPM at 60MPH.
This is the 1280 marked on the speedo dial in very small print, just above the odometer. This is on a MK1 S with the UK built Smiths speedo.
An OZ MK2 Australian built Smiths speedo I have has 1312 marked below the odometer. Standard diff ratio in both cases is 3.44 and wheel and tyre size is also the same. This is 2.5% more than the MK1 so I'm not sure what is going on here as the difference doesn't seem to match any standard ratios, although the internal pawl and ratchet set up on the MK2 odometer is different to the MK1 - metal pawl and ratchet gear on the MK2 vs plastic pawl and ratchet gear on the MK1.
Later UK built cars changed the speedo drive ratios and presumably used a standard internal gear speedo head across the range.

This is one thing our American friends got right.
In the USA, 1000 Rev Per Mile is also 60MPH at the speedo head and was the standard arrangement. Chrysler did this and presumable the other US makes did the same.
It was much easier and cheaper to match the gearbox speedo drive to give 1000RPM at 60MPH whatever diff ratio and wheel and tyre size was used rather than changing each speedo head to suit.

If you change back to the original 3.44 ratio vs the 3.1 you will get slightly better acceleration and lower max speeds in the indirect gears but cruising at 60 MPH will be at higher engine revs and noise - roughly 3800 RPM vs approx. 3400 RPM.
You also need to find out if the speedo head has been altered to suit the 3.1 59/19 ratio.
The speedo needle can be easily repositioned to show the correct speed but the internal odometer gears need to be changed to show the correct distance although this doesn't really matter.

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68 build Mk1 Cooper S
72 Clubman GT project. Total restoration required

My Understanding Of the numbers like 1280, 1312, 800 820 etc usually found in small print at the bottom, usually near the odometer. Is that this is the revs that that Speedo requires to read 60 this can be 60 MPH or 60 KPH

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" A Hundred Years From Now It will not Matter what my Bank Acount Was,
The Sort Of House I Lived in, Or The Kind Of Car I drove ... But The World May Be A Better Place
Because I was Once Important In The Life Of A Child"

John, the kmh ones are rpm at 100 kmh.

I run a 140kmh Moke one which is 800 turns/km, it is pretty close for a 3.44 diff.
Equivalent to 1296 turns/mile.

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DrMini- 1970 wasaMatic 1360, Mk1S crank, 86.6HP (ATW) =~125 @ crank, 45 Dellorto (38 chokes), RE282 sprint cam, 1.5 rockers, 11.0:1 C/R.

With a diff ratio of 3.105 and Speedo 1312 and Mk2 gear ratios then when speedo says 40MPH I am actually doing 30MPH.

Have I got this right?

If you put all the details in here: http://www.guess-works.com/Tech/ratio.htm

165/70R10 tyres
Mk2 gear ratio (although this doesn't affect the speedo)
3.10 diff
1312 TPM (turns per mile) speedo
6/17 standard speedo drive gears
At actual 30mph the speedo will read 26

If you have
165/70R10 tyres
3.1 diff
1312 speedo
The suggested gears are 7/17

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59/19 is probably from a later model UK mini, as I don't believe it was ever available in an Aussie buil Mini.
And the later UK Mini's used different speedo pinion and drive shaft in the gearbox, along with the odometer gears to get the correct odometer reading.
So I believe you will need to change the speedo pinion and drive shaft in the gearbox to the correct UK one (there had a few ratios), and may need to change the gear on the odometer cluster in the speedo to get the correct odometer reading. This later part assumes the parts from the later UK odometer cluster will fit the Aussie one.
Then the speedo can be set up to give the correct reading.
Aternatively you can live with the error in the odometer reading, and adjust the speedo by changing the needle location as someone mentioned above. - this is a rough way to change the speedo reading and will not be 100% accurate at all speeds.

timmy201 So where can you buy the suggested gears?

If you use the guessworks calculator it'll give you a link to find the right part numbers

7 tooth worm gear is DAM6028
17 tooth pinion is 22A1881 or DAM2904

I would want to be 100% confident you've checked all the parameters before trying to track these speedo gears as they are hard to find. Is your speedo labelled 1312?

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My speedo is 1312 & running 145 80 10 Nankangs.