As Tony says, many cars have a common fuse for the radio and cigarette lighter. (You might also find that the interior light isn't working, as that's commonly on the same fuse too). I had to replace the one on my Escort just a week ago. (It was 15A though, not 10A). The handbook should tell you where to find the fuse, how to remove it (with the fuse puller that's clipped inside the fuse box) and the correct rating. I paid just £1.89 for a pair of fuses at Halfords, so it shouldn't be an expensive job!
Remember tough that fuses don't blow without a reason. You've probably inserted something into the cigarette lighter socket which shorted the circuit. (That's what I did). So take a look at anything that you plug into the socket to check that it's not defective.
A new remote key will be far from cheap wherever you go for it. As Tony's link indicates, you'll probably have to fork out a couple of hundred quid (or simply continue using the manual key!)