I am running a set of lowered Protech shocks which have a fully extended length of "Lower (F 303mm Open - R 347mm Open)". My Rover Mini is running the same tires. Everyone measures from different places but I tend to use the bottom sill line. The measure the fronts and rear right before it curves up to the wheel arch. With that, mine measure 8.75" in the front, and 9.25" in the rear. I am also running the MiniSpares Evo cones (not sure if they are shorter) with HiLos, naturally. With that, I still needed to install a limiting strap on the rears to ensure the suspension does not droop too much, allowing the cone and trumpet to not be under any load. 

 

Looking at the numbers posted above, it appears these shocks have a much larger fully open number than the Gaz, which is likely why I need the limiting strap. Shocks are wonderful though.

To answer your question. I would say 1 inch in front, 1 1/4 inch inch at rear. 

 Calver's shock length list https://calverst.com/technical-info/damper-lengths/
shows that the stock KYBs (Kayaba in the UK) are as short as most of the "short" shocks. If the chart is accurate and adjustability isn't necessary they look like a good option.

Kelley

The front mini suspension has a full travel range about 3 inches. You can lower the mini an inch and then modify the top shock mount up an inch so the shock does not bottom out. You will not have to buy lowered front shocks. If you lower the rear, trimming the trumpets beyond the ratio for the inch, you run the risk of tire rubbing the body when cornering. Rear suspension travel is more than the front.

If you go by the numbers on GAZ shocks, lowered type fully compressed is 215mm fully extended is 292mm. Stock length for GAZ, fully compressed is 222mm fully extended is 318mm.

I had my measurements from the ground to the wheel arch when I was fooling with the ride height, I'll look and see if I can find what the numbers are on my set up. I have standard length shocks, standard cones and HiLos set so I only occasionally bottom when I'm really loaded up or driving "in a spirited manner"

I've found with my car (165/60/12 tires) you'll hit the tire in the wheel well in the rear before the standard length shock bottoms.
In the front the geometry gets weird when you go too low even with adjustable lower arms.

Also worth mentioning is speed bumps become challenging when the car gets low enough that you'd need shorter shocks.

There are front upper shock mounts that are made with the mounting arm raised so you can run regular shocks on a lowered car. Won't help with the rear though. https://www.minimania.com/part/C-AJJ3359/Classic-Mini-Upper-Front-Shock-Mount-Lowered-Heavy-Duty-Pair

How far can you lower a Rover Mini before you need short shocks?