Any 'active' TV channel (which is being recorded or watched) requires its own tuner circuit. A standard Sky box only has one tuner, whereas a Sky+ box has two. Adding a separate DVD recorder will provide at least one extra tuner circuit but, since it won't have a valid Sky card associated with it, it won't be able to receive the encrypted subscription channels. That means that, without Sky+, you can't watch one subscription service while recording another. (It doesn't matter what make or model of DVD recorder you've got; it simply can't be done).
However it is possible, by using a suitable DVD recorder (or PVR), to record one Freeview or Freesat channel, while watching a Sky subscription channel. (The tuner in your Sky box will handle the encrypted subscription service, while the tuner in the recorder will handle the free signal). However you'll also need one of the following:
(a) a 'normal' TV aerial on your roof, with a coaxial cable to the recorder if you'll be using a Freview recorder ; or
(b) a dual (or quad) LNB attached to your satellite dish, together with a separate down-lead between the LNB and the recorder, if you'll be using a Freesat recorder.
So we need to know whether you want to use Freeview or Freesat for the service used by the recorder.
Chris