When you cancel Sky then the Q box and 3x minis go back to them. That means unless you have cables going to the rooms for the minis, then you're either going to have to rely on Internet catch-up TV or have some cabling installed.
The catch-up services are available right now for you to try. Use a smart TV or tablet/PC to give then a whirl. Personally? I think the functionality is clunky and the quality pretty variable, bur you might be happy with it.
Bear in mind that smart TVs have a habit of ceasing their support for apps after the first 3 years of a TV's model life. For this reason if nothing else, I would definitely consider putting in a more robust backbone. Your options are either Freesat or Freeview.
Freesat requires single or twin sat coax to each room depending on whether you want to watch live or record. With four rooms to service you will also need a change of LNB to an 8 output version. The current Q LNB won't work with Freesat gear.
If it was me, I would have an aerial system installed instead. One wire to each room enables the existing TVs and it's still all you need even if you want to run a dual channel recorder as well as watch live on the TV.
Add a Now TV box + subscription for the main TV only (£20/month) to get Sky Atlantic and some of the other entertainment channels not on Freeview and that should cover you and give a saving of £80/month.
Beyond the one-off cost of the aerial system (budget £300-£400 roughly), then the only other hardware cost is for any additional recorders per room. There's plenty of choice. At the cheaper end of the market you have the Chinese boxes sold via Ebay. In the mid market there's the budget boxes from Bush etc sold through Argos and the like. At the upper end you have Humax with quite slick machines which include decent support for catch-up TV.