Your PC is capable of buffering streaming services for you (running Netflix on your PC) but a standard TV isn't. A Roku buffers but doesn't have as much storage capacity to do it as well as a PC. With poor broadband speeds, I suspect that there isn't much available to you. I think I'd nag my broadband provider a bit to see if they can make any improvements. You said you are with Sky (broadband provider?), I can't find the link but think I read somewhere that they can boost broadband for ‘gamers’, it may be worth asking them.
You could also try using a wired connection to your router rather than WiFi which may also improve speeds, but you would have to connect to the ethernet port on a Roku 3 (I think, but you'd have to do your homework, I haven't tried it).
I have a Roku with my non-smart TV and it works fine, I also have a smart (stupid) TV that doesn't have Netflix and I can't add it, so if you change your mind and get a smart TV, check carefully which apps are available.
I would think that something like an Echo would cope with streaming radio as data streams are much smaller.
Good luck with this, poor broadband speeds are often the biggest hurdle and mean that your options are very limited if you want to use streaming services.