I should say that I want to use vga here even though there are hdmi inputs to the tv available; they're used for satellite, dvr, etc and i can't be asked to faff about swapping back and forth or switch those on and off to move to pc. I didn't have to do that using the available vga input, so I want to stay with that.
Just to ensure your question is related on connecting a HP 260 G1 with an Intel HD Graphics ('HD4000'). But you are trying to connect the system via VGA to TV. And only getting a blue screen. But an normal monitor it works fine.
First let me ask to ensure we are on the same channel. Are you connecting the computer directly to the TV Set via a VGA to VGA cord or what type of connection.
Also you mentioned that the TV Set also has HDMI ports and are been use by other devices. With this I will first recommend to check with the manufacturer of the TV Set to see if the VGA port might be disable or problems when any of the other input ports are been use. Also to verify the Resolution the VGA port on the TV Set is able to handle as Max and Min.
This is because to save Hardware the manufacturer of TV Set will share the same communication port for two or more devices. Also if the VGA ports uses an exact resolution this will be need to be configure on the display properties of the graphics.
Last but no least I will recommend to check with HP to see if they have encounter any problems with VGA connection.