£300-£400 for a 40"-50" TV isn't quite bottom-budget money, but it's not far off.
You'll be tempted (and possibly advised) to go for the biggest screen you can afford because "
you soon get used to it" or "
you'll wish you went bigger". There's truth in both statements, but it leaves out the important fact that this applies far more to TVs in much higher price categories where all the essentials and a lot of the
nice-to-have features have already been covered. That's not your situation. With the money you've earmarked, you're still in the territory of
which-essential-can-I-afford-to-sacrifice.
Where you plan to game, and you're going to the expense of purchasing the next-gen XBox to do it, then one essential you definitely want to have is low screen lag. If you're not sure what this is, it's the time delay between you pressing a button on the controller and your FPS character or car etc reacting to that command.
The best way to get low lag is with a TV with a high-end LCD panel that can display more images per second than the standard 50 or 60 fpor most TVs. It's not the complete solution, but it's a big part of it. This is called the Native Refresh Rate. It is not to be confused with the electronic picture processing sometimes referred to as the motion processing rate. As a very rough rule of thumb, you'll often see that the motion processing rate is double that of the native refresh rate. i.e. The motion processing rate is 100/120Hz, but the panel itself is only capable of 50/60Hz.
TVs with these faster panels generally start to appear at double your budget, so you need a second way to get low lag. What you should be looking for then is a TV with Game Mode. This switches off as much of the picture processing as they can get away with so that there's as simple a signal path as possible. Not all TVs have this, so if gaming is important, and your budget means you're looking at TVs with 50/60Hz panels, then ask or look up the feautures in spec sheet.
TL;DR version
For your budget, buy a smaller set to get a better chance of higher performance. Avoid the really cheap models such as Bush, Toshiba, Hitachi, Linsar, JVC etc. Only buy a major brand such as LG, Samsong, Panasonic, Sony if you know for sure that the specification gives you all you need. Most of these brands only have a token presence in the budget arena. They save their 'good tech' for their higher-priced models.
I'd recommend you have a look at the
Hisense 43AE7400. This is available for around £380. You get 4 HDMI 2.0 inputs (that's very good) and means there's plenty of input capacity for your ATV box and other sources. There's Alexa compatibility too. The input lag is a decent <20ms according to the specs, and there's a full compliment of HDR formats: HLG, HDR10, HDR10+ and Dolby VIsion (DV). It even handles DTS and DTS:X audio which is a feature sadly missing from some higher-end TVs.
