Your blackmoor had whitespot from the shop and did not get it from being moved. If the fish was treated successfully there won't be any whitespot parasites left in the tank.
If you leave the new tank without fish for any more than a couple of days you will need to add ammonia to keep the bacteria going. Without a food source the bacteria will go dormant and you will have to cycle the tank again.
If the tank has cycled and the blackmoor is going to be rehomed into that tank, then move it in today so the filters stay active. You can use the other tank as a quarantine tank and use it to house any new fish you get. You keep any new fish in the quarantine tank for a month and if they have been good during that time, then add them to the other tank.
If you leave the new tank without fish for any more than a couple of days you will need to add ammonia to keep the bacteria going. Without a food source the bacteria will go dormant and you will have to cycle the tank again.
If the tank has cycled and the blackmoor is going to be rehomed into that tank, then move it in today so the filters stay active. You can use the other tank as a quarantine tank and use it to house any new fish you get. You keep any new fish in the quarantine tank for a month and if they have been good during that time, then add them to the other tank.