B&W
I'll just cover the WIFI side, so this assumes you're only buying access points, since they will work fine with your Sky hub.
You basically have many options on how your Unifi network will function.
This splits two ways, you can run with a controller or without a controller.
Controller - What this does is gives you a web interface that lets you change every setting on your wifi devices / network.
https://demo.ubnt.com/
This shows you how it looks^
I'll start with the cheapest and move up..
1. Mobile phone application.
-You don't bother with a Unifi controller and use the phone app to quickly set up each Unifi access point. Up and running in minutes. You can use the app to upgrade your APs and set the SSID etc, simple. This is what I use when installing one or two APs in someone's home.
2. Windows / Linux / Mac controller.
-You install the controller on your PC, so your PC needs to be on with the controller launched for you to access stats and make changes to the WIFI. The WIFI will function perfectly fine without it being on, unless you want to have a guest portal etc.
3. VM controller / NAS controller
-Same as above really, but you contain it within a virtual machine either on your PC or your NAS (if supported)
4. Raspberry Pi controller
-Get a Rpi3 and install Unifi on this, cheap enough to leave on 24/7, will collect stats and accessible all the time
(this is what I use)
5. Buy a Unifi server (£1000+)
-lol
6. Use a Unifi Cloud controller
-Many retailers offer a 3 year free cloud controller, it's a few £ per year per access point after that.
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It's worth pointing out that the APs are PoE. This means that using either a PoE network switch or a PoE injector (they come with these), you can inject 48v into the ethernet cable that feeds them, so you only need a network cable at one end to power them up.
You can use an 8 port Unifi switch for this (US-8-60W) £100 ish, or a non Unifi PoE switch £50 ish. The Unifi switch will give you stats.
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The USG is a router replacement, that gives you more stats again, I'm not familiar with Sky's set up, you may need a modem between the wall and the USG.