Eliminating sugar will help you to lose belly fat for two reasons.
Firstly, and most importantly, is simple calories in vs calories out. To lose belly fat, you need to lose global fat, over your whole body. This means maintaining a caloric deficit (500 calories or so per day to lose 1lb/0.5kg per week). As each gram of sugar contains 4 calories which, realistically, do not fill you up or bring any longer-term satiation, it is a good candidate for cutting when reaching for this deficit.
Secondly, cutting out sugar will deplete your glycogen reserves. This means that your body will have to burn body fat for energy, thus depleting your overall fat reserves.
There is no way to spot target fat reduction, but by bringing down global fat reserves, you will rid yourself of unwanted belly fat.
You should expect to see results as soon as you cut out the sugar. Though you may feel a little sluggish at the beginning (as your body adapts to more using complex carbs and body fat for energy), you will see your body fat dropping from week one, as long as you keep that deficit.
How to form good fitness habits #10- set evaluation points
You need to set up evaluation points throughout your training cycle. Of course, you should already have your overarching goal (preferably a SMART goal of some kind, like increasing your bench press by 20kg in six months or losing 6kg in 3 months). However, within these goals, you want to have regular points at which you step back and track your progress.
Let's take our 6kg weight loss in 3 months. We may be wanting to lose 2kg per month as an even split. This means setting an evaluation point once per month, at which you weight yourself and see how you are coming on. If you've lost 3kg, great- either continue as you are and hit your goal early or eat a little more to slow down the weight loss. If you've only hit 1kg, you know to reconsider your diet and/or training- increase your caloric deficit to bring yourself back on track.
You can, of course, take this further. Try weighing in every fortnight, aiming to see a 1kg reduction each time.
Without these evaluation points, you're sailing blind. You could jump on the scales in six months' time and see no change at all, where you could have tweaked things early to make sure you're doing what needs to be done.