When you say 'dead wood' do you mean the inner branches were not leafing up?

If the branches you cut out really were dead, i.e.there was no green showing under the bark, and if there is still plenty of other healthy growth, then I think it should be ok. I do hope so, they are very pleasing little trees.

Hopefully you only 'cut it right back' to the right side of the graft point at the top of the stem - if you cut beyond the graft, i.e., into the central stem, any growth you get will be off the rootstock in the ground, and won't be 'Kilmarnock' willow.