You know,,
For years and years I have been connecting peoples VHS recorders to their TV's and telling them how to record etc,, and the single biggest misconception of all, is that the recorder wont work if the TV is off... I have tried no end of times and wit, to convince people that the recorders work independent of their because they have their own tuners built-in etc etc.. This usually just goes in one ear and out the other... However, since I have been into the world of DvD recorders, Sky Digital and LCD TV's with HDMI etc, I am not so sure now, as the connections get more and more complex - Smart connections this and that - have got me lost mainly because a lot of the stuff which applies in one country, doesn't necessarily apply in the next, because of different levels of digital acceptance, so sometimes I see terms and names which make me simply scratch my head in wonder...
All I can say is, it depends on how she has the recorder connected.. If the recorder has a direct A/V feed from a Sky decoder etc, it will (at least - it "should") record regardless of whether the TV is on or off. If, however, the Sky feed goes straight into the TV then from the TV output to the recorder, voila,, it can't record if the TV is off, can it?
Given what I said, above about the complex connections available today amongst appliances, I stand to be corrected...
Hope you sort it out for her... Sometimes, little old ladies (not that your mum is necessarily "little old") can be the very worst to educate in things electronic...
Good luck...!!!