Tvheadend

You can certainly use tuners on a remote device (either using a second TV Headend instance or something like MumuDVB or DVBlast) However I'm not sure about waking up and going to sleep when you need recordings to start/stop.

Thank you - I suspect that I might be going down the route of over-complicating my system.

Maybe I should keep my system as it is and accept that I shall have to leave it on all the time (and perhaps consider upgrading components with more economical parts, if energy consumption becomes excessive).

I have just come across something on another forum that suggests that you might be able to use a Raspberry Pi to wake up a server (by WOL), whenever it is time for a scheduled recording.

It would involve translating the scheduling format (on the server) to cron format on the Pi.

Has anyone tried this or can comment what might be involved?

The OS isn't running applications when it's sleeping so it isn't possible for TVheadend to somehow let the OS know that it needs to wake up because a recording is scheduled, because TVH isn't really even running.
Have you considered scheduling hibernation for blanket times of the day when you generally aren't recording, rather than specifically for each program? E.g. All of the programs I record are aired between 4pm and 1am. So I have scheduled the server to sleep from 1am to 4pm. This is easier to do. It's not clear why you want to sleep the device, is it for energy saving or hardware wear and tear or noise abatement(or maybe all 3)?
If you go down the raspberry pi route, it might be possible to send a 'wake on LAN' token to your server so that it wakes when a recording is about to start but I haven't personally tried it. The setting is in Configuration -> Recording ->DigitalVideoRecorderProfiles Then in parameters, enter a command to launch the script in the 'Pre-processor command:' textbox. On the server side, you would have to configure your server to respond to a wake on LAN token. The question is, will the Pi wait long enough for the hardware to wake without complaining or aborting the recording? I can't answer that as I haven't tried it. It also sounds like a lot of work too!

My main motivation was really for energy saving - I calculated that my server is only ever being used for about 50% of the time, so it seemed rather wasteful to have it running for the other 50% unnecessarily.

It is a clever idea to have the server hibernate for specified times - unfortunately, my son likes to record kiddie programs that are often on during the day, so it would be difficult to pin-point a windows when there is unlikely to be anything on.

The consensus, from various forums I have since read, seems to be that if it is a server, just leave it on all the time and take the hit on the electricity. This being the case, I am currently considering whether it would be worth upgrading to a new generation integrated passive motherboard, which shall use much less power - something like a J4205.

Yes, I have to admit, until recently I have left my server powered on all the time. If you can identify a window maybe at night when you can hibernate, that could help. Also, it's possible to power down idle drives even when the server is powered up, which might help.
Here are some power saving tweak ideas for Ubuntu but you may be able to apply similar to your OS.

https://wiki.ubuntu.com/Kernel/PowerManagement/PowerSavingTweaks