It is possible to gently colour bleached willow. The only trick is really to keep it from looking patchy.
Either use something like shoe polish in the turps/white spirits/meths used to make the restoring mix, tiny amounts of shoe polish I mean or dissolve a bit of a child's wax crayon in the stuff too.
Woodworking stains do work, but they rarely match the original colour tone accurately. For some reason they seem to stain yellow, red or blackish.
Try it on a hidden bit first would be my advice.
To be honest, old ones just cleaned up and given a careful nourishing, look 'right'
I know someone who boiled up a load of willow bark and then soaked a scrubbed basket in the warm liquor for a while.
It darkened the faded willow really well, just very, very dark, iimmc. It didn't dry out as well as she hoped either, and ends that had been woven and crushed a bit too firmly cracked off later on.
I think you have to look at each basket and assess it before doing something like that to it though.
My first suggestions are tried and gently effective and won't damage the willow/hazel/cane, just refresh it.
Bound to be someone on the forum who deals with antique restorations, rather than just my practical basketry upkeep type knowledge. It'd be interesting to hear what they say.
M