PlayStation 2 launched in Japan in March 2000, and most other major regions in October-November 2000. However, it's important to note that the PS1 was still doing very well. I tried to look for good sales figures (someone respond if they have the exact numbers), but I'm pretty sure PS1 hardware sales peaked in 2000. PS2 had hardware shortages and many considered the launch titles and launch window lackluster. How would launching the PS2 a year later in respective regions have changed the console? And would the PS1 have sold quite a bit more?

Bonus: How then would the PS3 launch play out? Let's say it would still be the same launch time as in real life. Would Sony still charge a lot for the console and cram in all the features that they did? 

It would have put its launch right up against the GameCube and Xbox instead of a year earlier which I do think would have given a bit bigger competition at least at first. Still think it would have won the generation easily but I think it would have sold a bit less

It wouldn't have changed much, other than maybe help Dreamcast gain enough footing to stay in the market. But I was there. PS2 was like some sort of unconquerable machine, even before it launched. The hype was unlike anything I'd ever seen, and likely will ever see again. So if it had launched in 2001 (speaking about the US here), it would have done much the same to the Xbox and Gamecube that it did launching in 2000.

As for the PS3, that is a completely different thing imo. Xbox 360 was a serious threat, and Wii was the console to own. Had PS3 held out and let the 360 get a two year head start... man. With the price tag, and lackluster interest that it had at the time of its launch? Who knows. Maybe Playstation as a brand dies completely out. I know one thing, it would have been a really bad idea.

PlayStation had solidified its place as the new emperor with the success (and its cultural validation) of ps1. Ps2 was king before it even launched. You had to be anti Sony to avoid buying a ps2. Some N64 and DC owners. But every ps1 owner and their closest friend was going to buy a ps2. As long as it didn’t change its strategy, it would have won the generation no matter if it was late. However, Xbox would have probably absorbed 20-30 million of its sales. Without ps2 in a place of market dominance, shopper would have compared its potential and specifications a little closer and that would have helped Microsoft.

I may have held off a little longer, It was my first non Nintendo console and I justified buying it for not only the console but the dvd player. A lot of dvd players cost similar or more at that time. It would have still won by a landslide but it probably would have sold maybe 10 million fewer.

More time for Dreamcast!

A General Electric DVD was about $99 back in 2001 and other brands around the same or $150. 

The PS2 launched at $299 and was the same price when GameCube and Xbox launched.

Just saying, because there is a premise of the PS2 selling a lot because it was the cheapest DVD player available. 

As has been mentioned, the entire retro gaming community would collectively cherish the extra year of amazing Dreamcast games that we would have been given. What I would give for a second Skies of Arcadia game...

I do not think that it would have changed much with regards to the PS2's success. The fundamentals of the PS2's success like the DVD player, Sony's amazing relationship with third parties, the dominance of the PS1 which was a gateway to the PS2 as well as the general weakness of the competition at the time would be unchanged. Honestly, delaying a year may have even made the PS2 sell a bit better since it would probably have boasted graphics somewhere between the Gamecube and the XBox just thanks to the faster hardware that would have been available with the extra year. The one complaint that I recall people having with the PS2 was that the Gamecube and especially the XBox had noticeably sharper graphics. I know a few people back then who went with the XBox just because it was the fastest technically and honestly and I really believe that games like Gran Tourismo 3 could have packed an even bigger punch if it was released on top-end hardware. I think that a few folks might have jumped ship from MS to Sony in that gen if Sony had something that was really top of the line for the generation, although it would probably just be a drop in the ocean in terms of the PS2's lifetime sales.

PS1 and Dreamcast would sell better. PS2 would be largely unchanged. However if PS2 is one year later, then PS3 probably is too. So much later compared to 360 and the Wii, the price would have to be even more competitive leading to more losses for Sony with the PS3, but maybe it would launch with some sick games and dominate, who knows?