I had broken my radius and ulna 3 years ago and had undergone surgery to fix them. They were removed last year. Can I start doing pushups now?

Yes. That shouldn't be a problem, but you should check with your surgeon.
Most likely, yes. Assuming you have no wrist pain and a normal/near normal range of motion, you can start doing push-ups. Start gradually such as 2 sets of 10-20 per workout. Since you are strengthening multiple muscle groups, i recommend a recovery day in between each session of pushups. Thus, 3-4 days of week should be fine, gradually increasing the number of repetitions. Discontinue/reduce if wrist pain develops.