convalescing: to recover health and strength after illness; make progress toward recovery of health.
e.g., He was due to spend the next eight weeks convalescing
I am currently convalescing and can't help you with your homework.
If I were to hear that someone was bedridden; yes I would understand that person to be sick and to be in bed, but I would also presume it was a serious illness, more so if the person is not elderly and frail. I would naturally assume that the person is unable to work for some time, perhaps weeks or even months. To be confined to one's bed 24 hours a day suggests something is seriously the matter. If this reflects the OP's case, than that suggestion is the most fitting and appropriate.
To be laid up describes someone who is physically (not mentally) unable to do any work. It suggests an injury, and therefore a temporary state, and implies the person will return to work shortly. It's a phrasal verb; it's colloquial and very well-known (at least in BrEng). If an employer were to hear this over the phone, they may show concern, but not be unduly worried.
To be convalescing suggests that the person is no longer ill, but that the illness was a serious one. It suggests that the person is too weak to return to work, and needs absolute rest. This rest does not necessarily mean the person is confined to their bed 24 hours but neither does it exclude it. It depends on the illness or the injury subjected. A person who is convalescing is physically (and sometimes mentally) too weak to perform any strenuous task.