Think of your connection to the gaming server as being composed of three parts: from your PC/console to the BT Hub; from the BT Hub to the "internet"; and across the internet to the game server.
The first part is entirely under your control. The best you can do is connect your PC/Console to the Hub directly with an Ethernet cable. If you are using WiFi then that first part of your connection could be subject to all sorts of interference - this is why the BT support staff ask "are you wired", it rules out potential issues in this part.
The second part is the one that is (somewhat) under BT's control. It is shared with other users and (if you're on a copper or part-copper) service subject to various technical limitations on achievable speed. This is the bit that's covered by BT's speed guarantee, and it's what their speed test will measure.
The third part is not "controlled" by anyone - it's a mish-mash of network links and routing tables governed partly by technology and partly by commercial agreements. There's nothin you can do about this part, and very little BT can do about it (at least on an individual basis).
So, literally the only thing you can do is ensure you have the best possible connection to your Hub, and ensure that BT is meeting their minimum speed guarantee.
