The Wii U is currently at 13.56 million units. We know that the Wii U is no longer being produced, so even the most optimistic projections leave the Wii U at 14 million, which is the amount that I am predicting the Nintendo Switch will reach by March 31, 2018.

The Nintendo 3DS launched at the end of February, while the Nintendo Switch is launching at the beginning of March. From its launch to the end of March 2012 (FY3/2012), the 3DS sold 17.14 million units.

During that time, these are the only 3DS games of note that Nintendo released:

Nintendogs + Cats, Pilotwings Resort, Zelda: Ocarina of Time 3D, Star Fox 64 3D, Super Mario 3D Land, Mario Kart 7, and Kid Icarus Uprising.

In the Nintendo Switch's first full fiscal year, it will have:

The Legend of Zelda: Breath of the Wild, 1-2-Switch, Mario Kart 8 Deluxe, ARMS, Splatoon 2, Super Mario Odyssey, and Xenoblade Chronicles 2.

And those are just the ones of which we are currently aware. Nintendo frequently reveals holiday titles at E3, so we can probably expect at least one more major game this year. Additionally, the Switch Fire Emblem is slated for 2018, and it could very well be before the fiscal year ends.

In fact, the Nintendo Switch fixes all the launch problems that the 3DS had: 

1. It has a constant stream of major Nintendo titles, including 3D Zelda, 3D Mario, Mario Kart, and Splatoon (which will be huge in Japan). It's clear that the Nintendo Switch already has a superior lineup compared to the 3DS during the launch timeframe.
2. It has a reasonable power to price ratio. Unlike the 3DS, which launched at $250 with a 240p resolution and less power than even a GameCube, the Switch is launching at $300 with a 720p screen, TV compatibility, and more power than a Wii U. This power leap is greater than two generations of Nintendo home consoles, and it runs Breath of the Wild better than the home console it was originally designed for. I think people are seriously understating just how impressive the hardware of the Switch is as a portable device.
3.  It has more coherent branding, Nintendo's marketing presence is much better, and the system's hybrid functionality is far more appealing than the irrelevant 3D screen of the 3DS.

For all these reasons, I believe the Switch will sell close to 3DS levels, if not better; i.e. at least 14 million in sales in one year and one month.

I hope this turns out to be true. I'll be getting my Switch at launch to play Zelda on the go.

Woah, the 3DS sold much better than I remember. That's around Wii level sales. I could see the Switch outselling the Wii U very quickly, but I dont think it will in 1 year.

The 3DS cleared just barely cleared above 15 million with an extra month of Japan according to this site's data ... 

I'll grant you point one but point two is where logic sort of goes off rails ... 

Depending on how you define "reasonable" in this context it could very well be that or if we were talking in "relative" terms, the compute power to price value proposition isn't exactly what I'd describe as competetive since the Switch only offers a fraction of what PS4/X1 slim delivers at the same price! 

Arguing in terms of price is not what I'd advertise since the Switch does not have an advantage there and arguing from the point of perf/$ is not a great idea either when the competition out there does that better. What you probably wanted to argue was that the switch offers balance between maximizing power/portability ... 

Power is also not the most important factor when people are deciding to buy the system, it's content and offering a platform with just a perf/$ advantage get's you nowhere when we take a look at the original Xbox, Gamecube (had a perf advantage and launched at same price as PS2 was at the time but failed) and N64 (the same thing happened only this time with PS1) ...

Point 3 is whatever since hardware branding doesn't matter much with Nintendo compared their competitors ... 

How long did it take for PS4 and Xbone to reach that number?

1.) I'm not using VGChartz data, I'm using Nintendo's shipment data. That's why I said 'shipments' in the title and not just 'sales'. If you add up 3DS shipments for FY3/2011 and FY3/2012, you get 3.61 + 13.53 million which is 17.14 million. https://www.nintendo.co.jp/ir/library/historical_data/pdf/consolidated_sales_e1612.pdf

2.) I wasn't comparing the Switch's power to price ratio to those of the PS4 or X1. Obviously a portable system has to make some sacrifices, and the Switch is great in the context of a handheld video game system. You'll note that at the beginning of that list I was comparing the Switch to the 3DS: It has a better power to price ratio than the 3DS at $180, and especially at $250.

3.) It mattered with the Wii U, which to this day is obscure among people who are not either Nintendo fans or serious fans of video games in general. Not to mention how the Wii U started off with a poor reputation because of its aesthetic and perceived lack of power. You may not think that matters much, but the Wii's branding (Wii would like to play) was exceptional, and definitely contributed to its success. The Switch is starting off on the right foot, just like the Wii. Even the 3DS had poor branding at the start thanks to its frivolous 3D screen and lack of games, and it still shipped 17 million by the end of its first fiscal year.

Tagging for Rol to tell us the answer.

It took the PS4 exactly one year to reach 14 million, and it took the X1 about 1 year, 9 months, and 1 week. Of course, VGChartz measures sales to consumers, so you can probably subtract one month from these lengths of time to compare against shipments.

Thank you! I'm just gonna guess it's going to be something between that. 1 year and 4 months or something like that lol 

I definitely seeing Switch passing Wii U LT numbers at end of 2018. but on 31. March 2017. is little to early, that just one year to sell more than 13.5-14m.