Cycling is often recommended for bone on bone/severe arthritis of the knee. But there are certain situations that may be undesirable depending on each individual’s knees. Avoid standing when riding, if you are riding on hilly terrain, learn to use your gears to get the ratio you need to maintain steady pedalling. And yes it is very important that you have your bike set up for your specific body, seat height, seat front to back, and handle bars. You should get the ok from your doctor in any case and you could ask him/her if they can give you advice on cycling. They may be avid riders and have more information than the average physician.
And swimming is great, but avoid the breast stroke. It is not mechanically suited for knee issues. You may want to look into water aerobics as well, it can be very therapeutic as well as very easy to adjust intensity and range of motion to avoid irritating joints with issues.
And I imagine you have been given a series of exercises to do on your own. There exercises are very important and you should continue to do them until your doctor tells you to stop or you experience problems when doing them. In which case you should discuss it with your doctor. The same goes for cycling, if you start to have issues, talk to your doctor. You don’t want to have an easy to fix problem become a hard to fix problem. Good luck.