Yeah, you probably won't see success in breeding these guys with that diet (you might, but odds are not in your favor) - various species of phytoplankton have been tried a number of times unsuccessfully in the rearing of different Nassarius species.
I know of three species of Nassarius that have been successfully aquacultured, two of which (N. vibex and N. obsoletus) were aquacultured using planktonic feeds (Phaeodactylum tricornutum and Nitzschia closterium) that were much more commonly available back in the 1900's, and the third (N. reticulatus) was cultured using a suspension feeder product called "Snow Reef" by SHG.
You can still find Phaeodactylum tricornutum available at a hefty price from UTEX, and you can find a number of Nitzschia spp. on the market too (though Nitzschia closterium is not one you'll likely find), but I'm not sure how well these feeds would work for N. distortus. Supposedly N. reticulatus may be reared exclusively on Nitzschia sp. or spp., but, to my knowledge, this hasn't been attempted again since it was first attempted back in the 1930's.
Anyway, if you're wanting to try rearing N. distortus, these foods (Phaeodactylum tricornutum, Nitzschia spp., and Snow Reef) are likely your best bets for successful rearing foods. I've heard it can take anywhere from 3-15 days for the eggs to hatch (this is likely dependent on a number of different factors), and it takes about 1-2 months for the snails to go from hatchling veligers to settled snails (the time depends on the species and the chemistry of the available substrate in your tank for certain; it likely depends on the water chemistry too, but I don't know this one for sure; it also varies from species to species, with the average time for one being 35 days, the average time for another being 45 days, etc.). So, the total rearing time for these guys is likely 1.5-2.5 months.
One important thing to keep in mind if you try rearing these guys, though, is that high algae growth in the tank may inhibit larval snail movements and result in them dying, so it may be important to keep nutrient levels/algae growth under control throughout the rearing process.
I'm not sure how long it would take for copepods to establish in the tank (it depends on a number of different factors - water chemistry, food availability, etc.).