I want to earn money online by freelancing web development. How many days will it take, approximately, to learn HTML, CSS, and Javascript if I spend 2 hours per day studying?

TL;DR: 100 hours or 50 days before you can find your first work. 3-4  months to become good at those 3 and proficient in PHP(Back-end).  Then you get stuck in an infinite learning loop which is not as bad as it sounds.

Long answer:
You can learn enough HTML in 2 hours, but that's not enough since you need CSS and Javascript to complement it. Here's how I did it back when I was sluggish but still wanted to learn web development:
1. Learn HTML
2. Learn CSS
3. Learn the very basics of JS

Points 1. and 2. above took me a few days, at probably 1-2 hours max a day. HTML and CSS are just too easy when you learn them. I'll move on to the real stuff later.

For point 3., it took me longer. If I remember correctly, I struggled a bit at first with it. I was able to learn the basics of PHP faster than JS. I think it took me a good 15 hours to learn the basics of JS. Note that I never programmed before. I started learning to programme with PHP after learning HTML and CSS.

So, if you want to get the very basics, it would take around 40 hours, which is 20 days at 2 hours/day. Let's put it at 30 days to get an excellent base of all 3 of them. HTML takes 2 days. CSS will take another two days. JS will take the most time. Of course, you need to learn by writing code. So, fire up your favourite text editor, write simple stuff and test that they work in your favourite browser. At the end of a month, you will have several .html, .css and .js files on your machine. You may have some simple stuff, and you may have some .css files with hundreds of lines. It feels perfect when you see you did all that, trust me. ;-)

Now, the real stuff is to apply them. Once you have the basics, you need to learn how to use them all together. That's the "hard" part as you have to build something. There is no way around this. Think of something, design a page and code it using HTML, CSS and JS. That's when you will realise how much you don't know and it's ok. You will learn more by doing. If you spent your first month on the basics well, your brain should already be wired to mix them all. Build something complex. Find a complicated website and try to rebuild it using your newly found skills. Once you make a dedicated web page, make it responsive. Learn it! You should spend about 40-60 hours on building more and more complex pages.

You now have about 100 hours of HTML, CSS and JS under your belt. It's time to go freelancing and put your skills to good use. The real learning starts here. Prepare yourself for a life-long learning process. It never ends. You will always find that you don't know anything and you will learn it so you can do the job. What the client wants, the client gets. There are things I never thought could be possible, and I managed to do them. If you dig deep enough, you should almost always find a way to solve a problem. While you're at it, pick up JQuery, Angular, Node, PHP or Ruby or Python or <any X> language and keep working at it.