Well your range is obviously ok and you can sing on pitch but what you need to work on is timing, diction and breath control. There is a lot more to it that you will pick up from any good course but those things you can actually work on yourself without any external help.
Listen to the originals/your favourites versions of songs such as "You Raise Me Up" and try to match the timing and pronunciation - not the tone or timbre as you want that to be unique to you.
What a good course will also teach you is how to listen better and I think from your recording that although you roughly know the words and tune to that song you hadn't really listened in depth to a performance on which to base yours.
Hope that helps and keep working on it - how long it takes to improve is entirely different for everyone so no point guessing that.
- Should just add that I am neither a professional vocalist nor coach but they are just my observations.