Thgere have been several questions on this issue lately
Miffy. Best advice seems to be wait a bit and see if new growth starts in spring . Even if the top is dead they do often resprout from lower down and if they do just cut off the dead above the new growth.

This is what Bamboo wrote earlier this evening to an identical question -

"I'm afraid your Cordyline has been damaged by the severe winter weather before Christmas. Do nothing now, just wait and see what happens - if, by April, the rotting and oozing has worsened, then cut it down beyond that point, to the ground if necessary. If it doesn't get worse, then just wait - it may shoot halfway up the trunks, or from the base. If you have to cut it down, it may yet regrow from the roots beneath the soil."

Thank you for all the help with my cordyline. I will wait and see what happens before resorting to chopping bits off, but it's very reassuring to know others have suffered the same damage - i.e. it's not my ham-fisted lack of care that has caused it! I have bags of enthusiasm but very little know-how, as I grew up in a council flat with a window box which was MY DAD'S and nobody else's!

Thanks again - Miffy x