Cylinder Sleeve / Liner 1275 , Sprite , Mg Midget , Mini

Any machine shop doing this work will know what to do - it's a very common thing for them. You could use the pistons you have with new rings, however, with the cost of fitting the sleeves you may want to go with new pistons and rings instead of just rings and the reason for this is because the sleeves cannot be fitted with the standard pistons after fitting. They will still need to bored to standard size (or any other size like +.020", +.040", etc.) so the machine will need your pistons and rings so they can set the correct bore.

I usually recommend fitting new pistons and rings any time re-boring is done and especially in this case where you're paying for sleeving, too.

the engine will be better off with new pistons and rings if you're going to the trouble of resleeving. Otherwise, I would find another block that doesn't require boring but just honing and then you could use the pistons you have with new rings. But finding a standard block that doesn't require boring over would be a trick!

Those pistons would be fine, or the higher compression types for a little more oomph:

https://www.minimania.com/part/21253-STD-EA/Classic-Austin-Mini-Cooper-1275-Pistons-Std-975-Compression-Sold-Each