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There are various options that can be used in conjunction with an NTL box - but all of these carry the limitation that it is the NTL box that's receiving the signal - one programme at a time. So you can record, or watch, or both, but only one NTL programme at a time. Never record one and watch another.
The obvious choices from currently available gear are the various makes and models of HDD-DVD recorder.
Or TiVo. Which is a HDD recorder which can capture from external sources (such as an NTL box). It's no longer sold here, but used ones are available. And it's almost useless without a subscription to its proprietary programme guide.
As you've rightly surmised, Freeview PVRs are limited to receiving and recording Freeview from an aerial.
I do know that certain makes/models of HDD-DVD machine have an infra-red "dongle" that can be used to change channels on certain set top boxes. I can't speak specifically for the NTL box, though. Perhaps a quick look at (say) the Pioneer website - look for DVD recorders with Hard Drive - and look for Set Top Box Control - and if necessary, ask them. These same models also have an EPG (GuidePlus) which should make the job easier.
I also know that some Set Top Boxes have their own onboard timer event feature. If the NTL box has this then, again, a suitable HDD-DVD machine can be automatically turned on and off by the Set Top Box (i.e. the reverse of what you are thinking).
I must admit to knowing diddly-squat about cable, living in the sticks, but you can now buy a Dreambox DM7025C twin tuner cable PVR. I'd lay odds on that being able to do what the OP wants if he were to learn how to set it up.....
As already stated by LV426 a used/reconditioned Tivo will do what you want with standard definition TV.
If you were happy to pay an extra £5/10 a month to NTL (forever) for TVdrive and aren't bothered about HD then a Tivo with cable adapter will be a huge leap forward for £10 a month (or pay the extra for a lifetime subscribed Tivo).