You've used the term 'free-to-view' in your post but your reference to the absence of C4 and Five suggests that you might be unaware of the difference between 'free to air' (FTA) and 'free to view' (FTV) channels.
The vast majority of the free satellite channels fall into the 'FTA' category. They're not encrypted and you should be able to receive them via any Sky box, even without a valid viewing card in it.
A few channels are 'FTV'. Those channels are encrypted, so they can only be received when a valid card is inserted into the Sky box. At one time C4 was FTV, but it's now FTA. Hardly any FTV channels remain; the only significant exception is the 'Five' group (i.e. Five, Five US, Fiver and their '+1' channels).
If you can live without those few FTV channels you definitely won't need a new card for your Sky box. You'll still be able to get all other free (FTA) channels.
If you require the 'Five' group, you might need to buy a new card. Up until recently Sky have been selling 'Freesat from Sky' cards for �20 (or �19.57 after the temporary VAT cut came into effect). However anyone trying to access the 'Freesat from Sky' website is now directed to the main Sky website, where a search for 'Freesat' produces no results. So it looks like Sky have abandoned their version of Freesat (which is not to be confused with the separate 'Freesat' service offered jointly by the BBC and ITV).
I suggest doing absolutely nothing for the time being. You should definitely be able to continue receiving the vast majority of free channels irrespective of any changes to viewing cards. If you suddenly lose the 'Five' group (which I actually think will be unlikely) it might be worth investigating alternative cards, such as the one linked to by Anotheoldgit.
Chris