A belated update...It worked, though there were a few minor issues.
I got hold of a Pi 1B+ and connected it to the Decoder SCART socket of the Amstrad analogue satellite receiver.
To get the receiver to display the picture coming in through the Decoder SCART (as opposed to the snowstorm being received by the tuner, due to us living in the 21st Century) I had to bridge pin 8 of the Decoder SCART with pin 8 of the TV SCART with a length of wire. This was quite simple. I then connected the TV SCART socket of the Amstrad to a normal television, tuned it to AV1, and switched on the receiver.
As expected, the receiver displayed the video signal coming in through the Decoder SCART from the Pi, no matter what channel I selected on the receiver. My little modification fooled the receiver into thinking 'ooh, this person wants to watch a descrambled signal coming out of a decoder' when in fact the descrambled signal coming our of a decoder was a VLC stream coming out of an RPi.
The tricky bit was getting the Pi to change streams when I changed channels via the Amstrad's remote. I tried tapping the remote IR receiver on the Amstrad's circuit board, but had no luck in getting the signal into a Pi or Arduino in usable form. In the end, I used an Arduino Nano with its own infrared LED to do this. The Nano interpreted the remote control signals (e.g. the hex code for 'button 4 pressed') and gave instructions to the Pi (e.g. 'change to stream 4'). It was very basic; only channels 1-9 were supported and all other buttons were ignored, except the useless 'Authorise' button on the remote which I programmed to shut down the Pi.
I mounted the Nano and its IR receiver carefully in the Sky viewing card slot of the Amstrad. This meant that it was *just* possible for the Nano to live inside the satellite receiver, yet still receive the IR signal from the remote control - provided I pointed the remote in exactly the right direction! It was a bit of a faff.
I then programmed all the channel names into the Amstrad, so pressing button 4 would cause the Pi to start streaming 4Music through VLC, and simultaneously, the Amstrad's 90s-retro OSD would show '04 4MUSIC' as though I was watching it via Sky in '92.
The only finishing touch would be to power the Pi via the Amstrad's unused LNB voltage output via the F-connector, with a suitable step-down transformer to drop the 12/17V to 5V. I think the current supplied would be just enough to run the Pi...maybe I'll give it a go one day.