Having used Japanese water stones and Japanese chisels for the past 30 odd years I would suggest checking out some reputable Fine Woodworking supply stores for your stones and lapping supplies. Woodworkers Supply is one, Woodcraft is another and Lee Valley & Veritas (very expensive) is a third. They may or may not have better prices than chef's supply houses.
I have always used my nagura stone BEFORE every sharpening and never as a stone flattening tool per se. The nagura creates a slurry (mud) quickly, with just 10 or 20 seconds of working the entire stone evenly; and as stated by BDL above, it's the slurry that does most of the cutting. For keeping my stones true I have always used "wet-dry" silicon carbide sanding sheets (the black paper) on a piece of glass about 3/16 inch thick or thicker if you can find a scrap somewhere. Stop into some glass supply or picture framing shop and ask if they could cut you an 8 1/2 by 11 inch piece from scrap...you may get it for nothing. They break custom cut glass all the time, they'll have scrap. Use the whole sheet of sandpaper and cover the whole sheet evenly with the stone.
I have to take exception with the abrasive grits for the screens and sandpaper being discussed here. I've never used them so I don't speak from actual experience, but I've never used anything coarser than 400 grit, and that's for the 800 grit stone (my most aggressive) and then finished it off with 600 grit. The yellow 6000 grit stone is so soft that I start with 600 and go up to 1200 grit to finish it off.
I know, I know.......a little excessive but hey, I've been making my living as a furniture maker and antiques conservator for over 30 years...so sue me, I'm a fanatic for perfect tools and edges.
Also, I never take my stones our of the water tray, they're constantly submerged. If you use them less than once a week though, I wouldn't bother with keeping them in the water constantly.
Here's the big time saver: since my stones get heavy use and wear much more unevenly sharpening relatively narrow chisels (as well as my knives) I give them 10-15 seconds of flattening every single time I use them before putting them away in the water tray. This is the way to spend the least amount of time having to flatten them and you only remove a tiny bit of material every time so they're always flat.
Sorry for the very long post.