That's not normal. I'm sure your doctor's told you that too.
My first thought is that it may just be a severe case of being out of shape, I think you might be able to push through it. You didn't specify how long you kept it up but I would expect it to take at least a few weeks, maybe a few months before you really get up and running.
I would suggest that you keep trying, and taking it slower. Maybe instead of running and cycling just walk/hike. I find it much more comfortable to hike.
It's also useful to keep track of your pulse. There's a type of training called zone 2 training where the goal is to keep your pulse in "zone 2". That's quite low intensity training, but it's also very effective and much less taxing. Zone 2 depends on your resting and max pulse but it'll be something like 130-160.
I can also recommend rock climbing, particularly bouldering is very approachable. And of course weight lifting if that's more to your liking. You don't have to do cardio, there are many ways to be active.
I have had some issues with nausea while weightlifting, towards the end of a session I would get really nauseous. However that's just me being out of shape, after a week or two it subsides and it just feels good. I like the feeling of sore muscles.
Also, for me, activity is essential. When I'm not active I fall into a deep depression where everything feels harder. When I'm in shape I feel like myself, I have more energy, motivation and discipline, life feels easier. I don't think it's like this for everyone but I do think everyone benefits from exercise. Even if you have to fight through some bad feelings to get going.