Once cells have been started there is no reason why a cell raising colony should be queenless, and it is often easier for the beekeeper to provide a separate queenright stock to finish the cells to the sealing stage.
In UK conditions a strong stock that is occupying two deep brood boxes is capable of "finishing" one or two frames of cells. If more cells than this are available, then rather than overloading one stock, use two or more... The reason being that a stock that attempts to feed more than it can really handle will inevitably under nourish, or abort, some of the developing queens.
Cells only need 24 hours in the "starter" colony and another frame of grafts can be put in the starter at the same time that the first frame(s) are transferred to the finisher.
When feeding finishing colonies to bring them up to strength, and whilst they are feeding cells, I like to use honey diluted with an equal quantity of water which I provide in a Brother Adam feeder or a Miller feeder. If I am raising a batch of cells very early in the season (or very late) I will use two frame feeders in the uppermost box instead of an overall feeder (this allows the insulated and un-ventilated roofs, that I use, to be immediately above the "activity" in the upper box). When feeding dilute honey in this manner... Take care not to feed too much at any one time as the feed can ferment (due to it's dilution). This is not much of an imposition as you will be making frequent visits to the queen rearing colonies anyway.
When handling cell frames that have developing queens ensure that they are kept upright and free from vibration or physical shock so that larvae are not displaced from the bed of royal jelly that they are lying in.
I like to set out my finishing colonies with the uppermost box having a central frame of developing brood that is not young enough for them to make queens from. Flanking this will be two frames of cells from the starter colony the frames that are to either side of this central group should contain as much fresh pollen as possible and developing brood. The carbohydrate content of the diet is provided by a Bro. Adam feeder on the top. The lower box, which is separated by a queen excluder, (or more usually a Cloake board) contains the queen and the rest of the colony's brood.
Pollen provision is much more important than most texts indicate and frames or trays of Taber troughs will enable the pollen to be provided within easy reach of where it is needed. I have done some experiments whereby I raised a small number (2, 3 or 4) of queen cells in mating nucs [future link] The special frames that I used contain troughs for both pollen and liquid feed both above and below the queencells so that the distance between point of use and point of collection was an absolute minimum.
If the colony is to be used for several successive batches the central frame of open, but not young, brood should be replaced, every three or four days, either from the bottom brood box or from another colony. Similarly frames in the upper box that have had bees emerge, and are now empty of brood, should be replaced on the same three or four day cycle with frames containing sealed brood from the bottom brood box or any other source.
The cells may be removed any time after sealing and in a "production line" arrangement and placed in individual cages for emergence in an incubator. I have not tried this personally, as I prefer to leave them in situ and put them in the mating nucs 24 to 48 hour prior to expected emergence. One reason for leaving the sealed cells in the finishing colony is that the break from intense feeding of cells allows the brood in the rest of the colony to be adequately fed and looked after, because they are going to be "looking after" subsequent batches of cells.