when to add queen during a split.

I would add the queen when I made the split. If you wait to add her the split will start making queen cells and you would have to remove every one, or they will not accept the new queen. The split will know it is queenless in less than an hour after it is separated from the colony, but it will usually take 12 to 24 hours for the new queen to be accepted. Let the bees eat the candy and release the queen, if this takes 2 to 3 days, all the better as this gives the bees more time to get used to the new queen before they can get to her.

When requeening a full sized colony, I like to introduce the new queen to a 4 frame nuc and wait until she has been laying for 2 or 3 weeks, then I remove the old queen with 4 frames to a nuc box, and put the new queen's nuc in the brood nest area of the colony. I make sure the queen is on a frame in the center of the nuc. I keep the old queen as a backup until I am sure the new one is accepted.

This issue has been addressed many times here if you would do a search.
The young newly hatched bees will accept a new queen readily than the older
field/nurse bees which tend to be more aggressive toward the new queen. So I would
put a few frames of sealed brood with some very young nurse bees into a new nuc. Then
brush in a few frames of young bees and hang the queen cage inside between 2 frames in the middle of the hive.
If you don't have a new queen then they will make one or a few usually.

Hi do your split in the morning take away the old hive with the old queen I always make sure I take it few miles away then the new spilt put it in place of the old hive an add the queen in the afternoon or the next day is fine cheers

I put my queens in right then. It takes them about 24 hours to eat the candy and let her out. I did 6 last year and 8 this year. All 6 accepted the queen last year and 4 of the 8 I've checked on so far this year have accepted their queens.

One correction. Last year my queen cages had orange caps. I left them capped for 24 hours then took the cap off and let them eat the candy. This year had no caps, but haven't had any problems.

>If you split off a strong hive, would you add the queen right away?

I usually don't add one at all and let them raise their own, but assuming you want to give them a queen, I would wait overnight. Any longer than 24 hours and they get pretty far down the road of raising a new queen and then they may dispose of your queen anyway...