Hi !
I am so glad your getting better and making a goal to get back in shape, its the most wonderful thing you can do for you health!
Both are excellent exercises for lungs, and why not do both! The only different is they target different muscle groups, Jogging, is more over all work out,- it get your legs, upper body.Cycling is more on leg power, thighs, and calfs, and arms.
Both of these exercises are 'aerobic' , meaning they depend on your breathing/oxygen for fuel, so matter what you will be breathing hard only if you push!!
I would recommend jogging, it is excellent for cardiovascular system and over all fitness.
Start slow and listen to your body, you will cough alot of junk up but keep at it.!
Move on to cycling and change up your routines, as you don't want to risk boredom.
Also drink plenty or fluids, especially with some salt,.
Hope this helps !