I bought some from ebay, some from Aliexpress. Depends on how quickly you want them as to how much you pay. Just got a motion sensor for under £10 in a week off ebay. You can only use Xiaomi stuff with the gateway. There is other projects out there if you dont want to run the gateway and other equipment, check out, Zigbee2mqtt: getting rid of your proprietary Zigbee bridges (Xiaomi, Hue, TRADFRI) if that’s what you are after. You just need an US to UK adaptor to run the hub in the UK, most sellers will sell you one.
Yes thats correct, its goes the gateway, and then through wifi over your router to HA, there is no real delay, nothing more than any other kinda of sensor setup from any company. A second or two at the most to work through a script and send the sound to whatever you need.