This happened to us last year.
We were told to change the number on the original luggage labels. Ensure however that the deck number is the same.
We also made sure the check in staff were aware of the cabin change, as was our cabin steward.
You get a large black felt tip, cross out the old room number and write on the new room number. That simple. Alternatively the porters at Southampton will have handfuls of blank luggage labels if you need more. You hand theluggage to the porters before you get to checkin. If you have competed the online checkin and have printed off one piece of paper called the setsail pass, which has a bar code in the top corner then you need do nothing more. Personally I would go in and print off a new one that shows the new cabin number. This piece of paper is all you need to check in at port,and there will be a separate line for those with this.
Have a good cruise - my other half and his son will be on board too!