How much time would it approximately take to loose around 20kg (44 pounds) with a diet, walking for 30-45 minutes daily and stationary cycling for two hours a day?

depends on start weight, intensity of exercise, calories in diet, daily lifestyle, and a bunch of other things.
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Yeah what Wishes said.

What you eat is key to losing weight. You can exercise until you're blue in the face, but if you're eating too many calories, and if you're not eating the right kind of food, you won't lose. Also if you're eating too few calories, you can stall out your weight loss.

If you are eating healthily, reasonable amounts, and getting reasonable amounts of exercise, you could anticipate 1% of your overall weight per week to be a safe amount to lose.

I am vegeterian and am mostly eating vegetables. Is that good?

Eating lots of veggies is good, yes, but it still depends on how much you're eating. And I know plenty of overweight vegetarians because it's possible to eat too much of things like pasta, eggs, cheese, dairy, sugar and sweets, etc.

I dont eat much and I eat healthy (at least I will start when I start excercising for real). I think the problem is that I'm lazy and I just play computer games all day.

So you aren't actually doing this walking, plus 2 hours of biking daily yet? That sure sounds like a hardcore workout, esp. for someone who is used to just sitting around playing games all day, IMO. That's almost 3 hours of exercise a day...I would think you would need to work up to this.

I did it in the gym, but I'm tired of going there so I stopped till I get a bicycle. I have been also lifting weights after those ~3 hours, so I guess I can take it. I started excercising about a month ago, but I became fat in the past because I had just been home. Understand? Sorry for my English...