Hi Angie and welcome to GoY. Cordylines (New Zealand cabbage palm) all over Britain have been affected by the cold winter we've just had. If you look through the pages of questions you'll see several every day asking the same as you.
Have a look at the trunk of your Cordyline. If it has rotten areas or is oozing gunk which could be orange or pink, you'll need to cut it off below the rot so that it doesn't spread.
Otherwise wait and see if it will regrow. This may be from somewhere on the trunk or from the root system. Don't give up hope til June or July, as after previous cold winters plants that had been despaired of came back from the dead. Good luck.

Beattie, been doing a bit of research on whether the rot spreads - apparently it doesn't, according to some experts, so the jury still seems to be out on whether you need to cut down immediately, or, as is best, wait till at least April.

Thanks for that Bamboo. I don't have any cordylines myself but there's a row of gardens not that far away from us with at least one Cordyline in each - and they all look OK. I noted that some - the ones on the corners of the block - have multiple stems where they've regenerated after a previous very cold winter, quite a long time ago to judge from the height of the trees now.