I'm curious, are bastards considered one of the smallfolk if their mother was? For example, if Jon's mother is Wylla or the fisherman's daughter (we know he isn't, but that's besides the point), rather than Ashara Dayne? If he isn't, then what about his children with one of the smallfolk?
A bastard's status depends on a few things. The first is if they're acknowledged (in some form) by their noble parent. Without that, they don't even get a bastard surname, which as GRRM says, "is simultaneously a stigma and a mark of distinction." A completely unacknowledged bastard will live and die a nobody peasant no matter who their father was.
The second is if they're taken in by their noble father. This will generally grant them a noble education (including martial training if they're male) and noble upbringing that they would not receive otherwise. This is how we get examples of Ramsay Snow, the Bastard of Bolton, Aurane Waters, the Bastard of Driftmark, Harry Rivers, the Bastard of Bracken, Rolland Storm, the Bastard of Nightsong, Daemon Sand, the Bastard of Godsgrace, and yes, Jon Snow, the Bastard of Winterfell. (Note "taken in" does not always mean at their father's castle; e.g. Larence Snow, son of Halys Hornwood, raised at Deepwood Motte.) The mother's status is not relevant to their noble upbringing -- see for example Ryger Rivers, one of the many acknowledged bastards of Walder Frey, whose mother was a smallfolk woman who milked goats, but Ryger has a presence in the Twins and even thought he could chide his father for being discourteous. (Walder told him otherwise.) But in general we know very little about these bastards' mothers, though likely most were mistresses or paramours (the miller's wife that Roose raped is probably a notable exception), and the odds are against them being particularly noble.
So to clarify from above -- these noble bastards aren't exactly smallfolk, but they aren't as high a status as their noble father either, and they have a bastard stigma either way. To take Jon's example, it doesn't really change his status if his mother was the noblewoman Ashara or the wet nurse Wylla, since either way, after Ned "took him in", he is just considered a noble bastard. Maybe some people might treat him slightly better depending on what they believe about his mother's status, but it's not nearly as big a factor considering everything else.
Now, I'm not quite sure what you mean by "his children with one of the smallfolk". But I'm guessing by "his" you mean Jon Snow, and are asking what if he never joined the Night's Watch and instead marries a smallfolk woman while still being considered Ned's bastard and never learning otherwise. Well, if he did so, his children would be legitimate Snows, unless they choose to change their name to remove the bastard stigma. Would they be smallfolk... *waves hands*... they'd be Jon's children, probably raised in Winterfell. The Whitesnows might be considered a minor vassal house, slightly below the Pooles and Cassels, somewhat above no-surname vassal smallfolk families like Hullen's, Farlen's, Gage's, and Old Nan's. If the mother was a noblewoman (minor or otherwise), would it improve her children's status? Maybe a little, especially if they had somewhere to live other than Winterfell... but on the whole it would all depend on Jon's own status in the household of Lord Robb Stark. Mind you, if you're asking what happens if Jon doesn't actually marry this smallfolk woman, then his children would be bastards, straight up. (And obviously.)
BTW and just FYI, acknowledged noble bastards, among other benefits, can have very good prospects in marriage, they wouldn't have to marry smallfolk. (Unlike poor Mya Stone, never officially acknowledged by Robert and not given a lady's upbringing; though even she has a knight from a minor house interested in her.) Walder Rivers, Walder Frey's eldest bastard, married a noblewoman from the minor House Charlton and has several legitimate children. Daemon Blackfyre, while an outlier in many ways, did marry a noblewoman of the Free Cities. And we have examples of a noble bastard like Tyler Hill, who thought he had status enough to romance a (married!) former queen, and like Daemon Sand who actually asked Doran Martell for Arianne's hand (and for all we know Doran only truly rejected him because of Arianne's secret beotrothal to Viserys).
