So will the daycare cloth diaper? How many you need depends on how often he gets changed. A 20 month old could go 4 hours in a diaper before needing to be changed or he could be like mine and drink so much water I'm pottying/changing every 1-2 hours. So I would say 6-10 diapers per day. For every other day washing that would be 12-20 diapers. You could also go in between with 15 diapers and see if you can stretch that to every other day washing.
I love BG OS diapers. They fit great, work great, and retain their resale value. You might be able to get the old version 3.0's cheaper some places as well as on www.diaperswappers.com. But if you don't like velcro you'll need to buy the new 4.0 version to get the snaps.
GroVia has become quite popular recently, I think largely because of their AI2 and disposable options. I personally would not go for their AIO's (where the insert is actually sewn in), but get the shells and purchase the snap-in inserts (AI2 system). Then you would also have the option of using the disposable biosoakers if that was more appealing to the daycare. And if the insert just gets wet you snap in a new one and reuse the cover. That way you can buy less of the covers (maybe 3-6 per day) and more of the inserts (6-10 a day).
The shells are about $17 and the inserts are about $8. One BG is $18. I always, always recommend getting a variety of styles and brands. So if it were me I would start out with a couple of each. Maybe get 5-6 BG's (they usually have a buy 5 get one free sale online), 3 Grovia shells, and 5 soakers. See how it goes, what you like, and then but more of what's working. Oh and if he's a heavy nighttime wetter you're likely to need a few doublers- I recommend hemp. There are also other brand out there that are great like Thirsites duodiapers, Happy Heinys, Fuzzi Bunz, Romparooz, Baby Kangas, and Knickernappies. But your cheapest option by far would be prefolds and covers and it would aid more in potty training b/c they do not wick (therefore he will feel wet as soon as he wets). You can get a dozen prefolds for about $28, covers for about $13, and 2 Snappiz (no pins!) for $7. I would recommend two dozen prefolds, 6 covers, and two snappiz.