From your speed test results your issue is not with the connection speed to the Internet but if you can't stream iPlayer and Internet radio at the same time it sounds like a Wi-Fi bandwidth issue.
Firstly have you changed the BT router settings to separate the 2.5G Hz and 5G Hz bands by assigning separate SSIDs? If not that should be your first step.
If you already did this and confirmed the devices that can use 5G Hz use this where ever possible (connect to the SSID for 5G Hz and not the SSID for 2.4G Hz) then you may be suffering from Wi-Fi bandwidth issues of your current router.
In this case replacements like the TP-Link AC2800 would help as they are designed for higher Wi-Fi bandwidth.
It's interesting to see that you're on the OLD Infinity 1 speed of 40meg download, rather than the current 52meg which your line is more than capable of.
I hope OPs issues were sorted... can I pick your brains on a similar issue?
I've been pulling my hair out for a while regarding the response ti es and download speeds via my HH5 to the point where I prefer to use the ethernet ports to work and play and watch tv.
Finally I had enouh and contacted BT to sort it out - most of the stuff I did, with their help, you've identified but the outcome was that my HH5 is properly misbehaving and needs replacing. BUT to be honest it hasn't played nice since day 1 so it might just be a duff unit or (perhaps) a not so great product for our household. Regardless, it's going.
I was about to hit buy on the Asus DSL-AC68U when I became aware that it isnt approved by Openreach?! not sure if this is an issue (if you guys use it then I assume not) so I nearly bought a netgear D7XXX box but something was niggling me and I realised that my HH5 isn't being used as a modem/router.
I have fibre to the house a bif white box with fibre going in and an ethernet cable going out to my HH5 (connected to the red colured port) - the DSL port isn't used. this must be the modem?
I guess this isn't normal because I haven't found many posts relating to it but it feels like i don't need a modem/router, just a router as my big white box is the modem?
With FTTH then I think you do just need a cable router and the Asus equivalent to the DSL-AC68U is the RT-AC68U which I'm guessing is as good as the DSL version which I have.
Thanks for the confirmation TimCurtis - i'm just looking at the reviews on the Asus now.