Can I use a diplexer to get both services down the same cable?
Yes.
A diplexer is a filtered splitter/combiner. Each side contains a filter that's designed to pass a specific frequency band to or from the common connector with a low insertion loss, while at the same rejecting frquencies that the other side passes.
So you need to get the correct type. You didn't say if your radio aerial is FM or DAB, but it doesn't matter because you can get a VHF/UHF diplexer that will combine the signals from a TV aerial and an FM or DAB aerial. You will need a weatherproof type if you want to install it outside.
This one should be suitable. And it passes DC to the TV aerial for powering a masthead amp, if you've got one.
You can use another diplexer at the other end of the downlead to separate out the TV and radio signals. It can be a second one of the same type, or an inline type, or a wall plate with a built in diplexer. If you want to use a wall plate,
this one should be suitable.
You could use a splitter instead of a diplexer in the living room, but a diplexer has a lower insertion loss (about 1dB instead of 3 to 6dB for a passive splitter) and the filtering in the diplexer will also help by attenuating strong out of band signals.
You can also get a triplexer that will combine signals from separate TV, FM and DAB aerials. There's a selection of diplexers and triplexers
here and a selection of wall plates
here.