What we have currently is a recently purchased tv with freeview available. A recently purchased separate dvd player/recorder, and a recently (one off payment) purchase of satellite dish with sky box with a card.
That's the bit I was after. But missing one little bit. Has the dvd player/recorder got freeview built in? Most modern dvd player/recorders will record either direct from the aerial or from a SCART lead and if yours is like this you should be able to record either analogue, freeview or Sky and also watch another program on either analogue or freeview while watching sky or analogue, freeview or Sky while watching analogue or freeview as for analogue or freeview the dvd player/recorder should be independent of TV and Sky box.
However SCART signals on some TV's automatic turns TV to SCART and you may need to press the "Input" or "AV" or "TV/AV" button on remote to return the TV to Freeview/Analogue.
What you should have is the aerial plugged into the DVD recorder then a patch lead from TV recorder to TV (You could also have Sky box in loop but no real need) Then from Sky box a SCART lead to DVD recorder then another SCART lead from DVD to TV. Plus a SCART lead direct from Sky box to TV. (Last item may no be required if aerials go through Sky box)
With Sky on and DVD playing on the TV pressing "AV" button should move from direct(freeview/analogue), Sky, DVD.
This is assuming the DVD player/recorder has a tuner built in then you should have no problem recording one program and watching another.
However I bought my Hard-drive/DVD player/recorder cheap as it had no freeview built in only analogue and although I can record freeview using the TV's built in freeview it is not easy and one has to be very careful to use the right SCART socket on the TV.
In fact it is rather a pain as one SCART on TV is in only the other is in/out and with wrong leads it is very easy to get a loop and very poor signals.
It is simply a case of swapping the two SCART leads in back of TV if that happens.
The Sky box also has slightly different outputs on SCART leads and again if you have a problem then swapping them over may help.
If you still have problems then there may be an option in using Phono plugs and adaptors instead of SCART and even HDMI connectors but need to know what your TV and DVD have on them.
My Panasonic TV instructions has a host of options shown as how to connect up and maybe your TV instructions also has something to guide you.
However it does seem from what you say something is not corrected right as you should have much more flexibility in what you can record and watch.
I did have a DVD recorder and found I needed to use the expensive re-writeable DVD's although it should have worked OK with write once seemed to have problems. However with Hard-drive/DVD recorder all those problems vanished and I record on Hard-drive then transfer to DVD first removing adverts so I get more on DVD as well. But it may be worth getting re-recordable DVD's if you have no hard-drive did cost 5 for £4 in Aldi but not bought any for a long time.
Do tell us what you find. It may help other.