There's nothing wrong with your connection in the screenshots you've posted. You and your friend both have almost identical results, with no packet loss and a very low standard deviation for almost all of the trace route, which indicates a stable line.
The only host with a potential issue is 184.108.40.206 which appears to vary between 30ms and 75ms, which might be problematic for gaming. A quick google seems to show lots of people have been seeing a similar thing with that host. It's worth noting that this isn't BT's problem, it's that router specifically. Your friend is also seeing exactly the same so all I can suggest is that he might be joining games on servers that don't pass through that node.
Swapping routers or cables on your end isn't going to help one bit, as your latency to all nodes up to that point is clearly fine.
I am just browsing the forums looking for some kind of ideas on this issue, and unfortunately there seems to be little, at least to my knowledge, a number of posts saying post this check and run that test and I know nothing of them unfortunately and do not have a computer at home to run them.
But I am having similar issues as yourself trying to onlne game with my Xbox one and smart hub, or even have my Xbox one on the internet. I have Infinity 1 and should get around 45mbps, however, my Xbox wired or wifi will often run at 0.5-2 mbps, occasionally will show 45mps but then I get lag and high latency consistenly, specifically on Battlefield one and GOW4. I have the constant symbol on screen (differs depending on game) showing poor connection and then lags, sometimes to the point of booting me out of games.
A secondary issue is that my Xbox will not connect to the internet, I have had could not get IP address, or can not get DCHP (or whatever it is) I have assigned a static IP, reset, rebooted, and nothing improves it. The other thing is that 100% of the time I will not get on the internet, but will connect to the router, if I have my xbox to instant on, Cortana is always telling me "sorry couldn't connect right now" or "I seem to be havin a problems" or similar. It is the worst gaming experience I have had, it was better when I lived remote with a max 2mbps standard line with another ISP.
My impression is that the Smart Hub simply does not work with the Xbox one as there seems to be so little that is in support or on here other than "buy yourself a third party router" which is often said. I just wanted to let you know you are not alone in your frustrations.
Just a quick update before I phone about leaving the game is still unplayable and have noticed even offline FIFA 18 is sluggish '' players being dragged'' when the Ethernet lead is connected or wireless is enabled.
When i connect to the internet at all on the Xbox the game becomes sluggish.
I know moving provider probably won't make a difference but I have to try something as 99% of the time the game is unplayable.