Just an update. After having the atomizer part completely covered in acetone (nail polish remover) for a day and a half, I tried it and behold! it works. The gunge that was the old dried perfume slowly came out of the base and left a little trail.
I ran a fair bit of hot water through it and it went from a jet to a fine spray on the second go. We filled it with a favourite perfume of Alison's and the only thing left to check is whether it leaks around the top a problem we had on 2 others we tried.
Thanks all for the ideas.