I think we need to be sure we understand the conditions in your bee yard and your objectives. As I understand it, the weather is rainy and cold, the main (is it the main?) nectar flow is getting under way, your main objective is to produce some honey and your secondary objective is to grow as you go.
If the above is correct, you do not want to reduce your colony size any more than absolutely necessary else you will compromise honey production.. What is absolutely necessary is to make the queen think she has successfully swarmed. Hence, pull her out and make a nuc. Unless the weather is unusually cold for the season, I don't think you need to put three frames of brood in the nuc. One frame may be enough along with some stores, an empty frame or two and enough bees to look after the queen and brood. Do not put any swarm cells in the nucs else you will defeat your purpose. It would be a very good idea to feed the nucs.
Do not split the deeps but, rather, either leave them intact or leave the strongest hive and combine the six remaining deeps into two three deep hives. Leave honey supers on. I would off all but two or three swarm cells in each hive, which will keep the bees happy until your purchased queens turn up. A few bees will look after the few swarm cells and a lot of bees will make honey, which will be surplus because there is only a limited amount of brood to consuming it.