They can be pruned back to about 18 inches, but its best to prune about a third each year over 3 years - if you need to do it all now for some other pressing reason, it should regrow quite well, but there is a small risk.

I agree with Bamboo but add that situation is important. It prefers light to heavy shade and will regrow slowly but reliable in shade. I have seen them in full sun (landscapers have a lot to answer for) and they look a totally different plant - small, pale leaves, lanky growth and struggle badly. So where it is is important.

The ones at Bodnant were cut down to a 6 inch stump a few years back and they are now fully regrown and nice bushy plants.
Oh and wear gloves when you do the cutting, the yellow sap stains your hands and is very hard to wash off!

The problem with your plan is that you will be robbing the plant of its ability to make food. So you will need to feed it (fork in and mulch plenty of organic material) until it has put on new foliage.

Thanks for replies. The Mahonia is in partial shade but gets a fair bit of sun in the morning and looks pretty healthy. I'll be planting a rose next to it so that will need all the sun it can get.

Bearing in mind what advice has been offered, I think I'll retain a couple of old stems with leaves and cut down the rest. Once regrowth has started and there's enough foliage I'll then remove the last 2 stems. Hopefully that should safeguard against disaster.