The short answer to your queries is Yes, you can definately bring your Xbox on a plane.

US agency TSA [Transportation Security Administration] mentions that passengers can choose to pack an Xbox in any of their luggage. Xbox or any other gaming consoles are considered harmless appliances. However, you have to take the console out of your bag and place it in a bin during the security checkup.

Similarly, an officer has the authority to ask for additional screenings in some cases. Also, an unopened Xbox in its original package isn't required lawfully to go through an X-ray test but might be subject to one if the TSA officer deems it a requirement. 

Meanwhile, the official website of the UK Government doesn't specifically include provisions relating to an Xbox. But the provisions about electronic items is somewhat similar to the US.

As mentioned above, an Xbox can be packed in either a carry-on bag or a checked bag. It is safer, though, to take it in your carry-on as the console could get damaged in checked baggage due to mishandling by airport staff. 

On the other hand, if the size or weight of the Xbox exceeds the airline's limit, you may be asked to put it in the checked luggage. Electronic appliances with lithium batteries are often required to go in hand baggage due to the dangers of catching fire in reduced air pressure. 

Although Xbox doesn't use lithium batteries, the controllers could contain them. Hence, while the gaming console can be packed in any bag, the controllers should be packed in a carry-on.

A personal Xbox is usually not subject to extensive screening. However, if you are carrying the device with intent to resale, the product could require additional screenings and taxes.