If you are also looking at SE16 then this might be the wrong part of SE1, but Colombo street gym has spin classes and membership is £20 a month, so it's probably cheaper to join if you are going regularly.
For SE16 try Seven Islands. You can get a fusion card which saves money off pay as you go classes.
Classes are more fun than you think. I have a bike, but I too am a bit put off long cycles in cold weather and an exercise bike is an alternative. However, they are so boring. A class provides a bit more interest.
What I can't understand is why only bikes have classes. No one offers a rowing machine class, but I'm sure that's just as good for you.