If you can access the Internet and apps on your TV it would appear that your BT Internet connection is working OK.
You should check to see if BT Parental Controls are activated by logging onto your BT broadband account via MyBT.
What "On Demand" TV can you not access?
Is it via an app?
Have you tried via a web browser?
I apologise in advance as I am no techie but I have the folllowing set up: Sat Dish (as I cannot get terrestrial TV- too remote where I live) connected by two cables to a Humax Foxsat HDR then HDMI cable to the TV.
The Humax gives me all Freesat Channels but only offers ITVplayer (I get the same 'connect to internet message if I try to load it) I have a Bose 321 system but have taken that out of the equation.
I cannot tune the TV to DVB or Anologue as there is no aerial but, as said, I get all channels thru the Humax by selecting 'HDMI1' from the TV (there is also HDMI 2 & 3, AV1 & 2 and TV) and using the Humax remote to choose channels etc.
From the TV if I select 'Apps', using the TV remote, I can access the internet, youtube, netflix and BBCiplayer plus all th eother preloaded apps.
Normally I would select 'TV' then Guide then home menu then On Demand & I used to get all the On Demand Services.
Even though I know I am connected to the internet the message comes up 'You need to be conected to the internet for this service'
Hope that helps??
I suspect that the message 'You need to be conected to the internet for this service' is just a generic message when there is a problem because you obviously have an Internet connection by virtue of the fact you can access every thing else that needs to be connected to the Internet.
Which model number is the TV?
I connected the Humax to the Hub with a cable and can now get ITV Hub on it (can't get others because of where Ilive) A friend has a couple of different routers he said I could have to try if that solves it so I will have done that by the end of the week & update In the meantime thanks for all the generous help & advice!!