Which variety?

Best time is after they have finished flowering (the catkins are flowers). They flower on new wood produced in Summer.

If it's not a pussy willow, I would prune it in the dead of winter, while the sap is down.

Hakuro Nikishi is the Species

All Salix produce their Catkins on wood produced the previous year. So you prune them after the Catkins have gone which gives the tree time to produce new shoots. This applies even to ones which are grown for their coloured leaves. the best colours are found on new growth.

It's not uncommon for Salix hakuro nishiki to never produce catkins - if yours doesn't, you can prune it now. Classed as Prune group 1 by the RHS, which means late winter/early spring, and only if necessary, which in my experience with this particular variety, it most certainly is after 3 or 4 years!

From the pictures that I am finding, the catkins of 'Hakuro Nishiki' aren't very ornamental, anyway.

Too right Tug - fairly insignificant things, they're really grown for their leaves.

Strange, ours does. Got them now in fact. Will be pruned in about 2 weeks time when they have gone over.