Around here (France) we don't really use cement board in private home bathrooms, mostly green ("waterproof with quotes") drywall. Yes cement board is better, but properly installed drywall works fine. It's even legal in some cases to use white (non "waterproof") drywall with a membrane, although nobody does it because it just screams cheap low quality work.
If you don't want to remove the drywall:
First install the faucet to check it lines up with the pipes, if you have to cut them to length or anything. It's much easier to correct any mistakes in pipes placement now rather than later. Make sure you take into account the thickness of the tile and thinset in this process. Is the shower high enough that you don't bump your forehead into it?
Next, unless your faucet comes with something special to do that, you need to plug the holes around the pipes. I use expanding foam, but that will fall to the bottom between the studs unless you first insert something to hold it in, for example some newspaper. Wait for it to dry, then cut the excess.
Get a bucket of liquid waterproof membrane and paint the drywall with it using a roller. It's the stuff that turns into a kind of rubbery compound when dry, check your local code.
Get the nonwoven bands that come with the liquid membrane and apply in the corners before it dries, then generously paint them with the compound, they should be well soaked and impregnated. Also make sure to coat the expanding foam with membrane so the holes around the pipes do not let water through. If you want to adjust the pipes later then you can leave the holes for now, but you'll need to keep a little bit of membrane compound to do it.
Now I see there is no caulk between the dry wall and the bathtub. It wouldn't be useful anyway because there is no point in making a watertight joint between a bathtub and a material like drywall which will absorb water.
In the ideal case, you'd want to remove the bathtub, then apply the liquid membrane up to below the point where the bathtub meets the wall. That way, when you put the bathtub in after the membrane has dried, then you can do a caulk joint between two waterproof surfaces (bathtub and membrane) instead of bathtub and drywall.
OK so let's suppose it's caulked and dried.
Next you take the leftover unwoven bands and cover the gap between the bathtub and the wall with them. So they will cover the vertical "lip" of the bathtub, and then the wall. The compound will stick to the bathtub, stick the unwoven bands to the bathtub and the wall, and impregnate the bands, so you end up with a continuous fibre-reinforced membrane that ends up INSIDE the lip at the edge of the bathtub. That's the whole point.
Finish the liquid membrane bucket, you're not going to use it anyway so might as well apply it.
At that point you should be able to take a shower with no leaks.
Then apply tile. Since there is a bit of a gap between the bathtub and the wall, and the tile should go over the bathtub lip, you'll need to use quite a lot of thinset. Make sure you take the thickness into account when checking the length of the pipes.
I recommend epoxy or acrylic grout for the tile joints. Much easier to clean, and they don't get moldy.
