First, it's important to understand that these were mostly communal fields, not the later 'enclosed' fields, which were a hugely different way of approaching farming. According to some scholars, in the 14th century as many as two-thirds of the population of England lived in villages or townships which held fields in common. These systems varied even across England, and had local differences here and there within that variation, so what follows is of necessity a high-level view.

These fields were not necessarily one big area, though. There's plenty of evidence to support a practice that agricultural historians call 'scattering', where small patches of land, distributed widely, were set to crops or fallow in particular years. These might neighbour fields of the other type, or meadows (often for hay, rather solely than grazing), or woodland, or wild land. It's not a particularly efficient way to manage land, though, so the reasoning goes that there were advantages to it, most likely that flood, fire, landslide, storm damage or other calamity could not affect all of one crop in any one instance.

Fallow fields weren't empty, though. In England, any ground left uncultivated will rapidly produce a crop of nettles, dock leaves, dandelions and so forth, as well as a wide variety of grasses. So there were sometimes used for pasture, particularly in common - being usually left fallow the year after they'd been used for a cereal crop. In many cases, though, it was required that a farmer place their sheep or cattle with a herder employed by the village, rather than have them as an independent 'by-herd' - this was to control the fallow areas grazed, so that none were over-consumed, or reduced to mud. Access to the land was very much a local decision and one based on current circumstances, so that the amount of time in which common grazing could take place could vary from one year to the next depending on conditions, as could the number of animals that any person or household could graze there.

People tried to sow crops in fallow fields sometimes; there are records of laws that quite explicitly ban this. From Sedgefield, Durham, in 1352 "This is an agreement by the neighbours of Sedgefield at the command and insistence of the steward and by consent of the neighbours, that one-third of the fields of Sedgefield shall lie fallow every third year and if anyone shall sow any part of the said third contrary to the ordinance the products of the same shall be forfeited to the lord and in addition he shall pay half a mark." That's undoubtedly been cleared up from the original language, but it's as quoted in Warren Ault's book (see below), and the intent is pretty clear - leave the fallow field fallow.

So, given that the fields were expressly not to be used for cropping, and were often used for pasture (which would have had the addtional advantage of adding animal manure), it doesn't seem terribly likely that they would have had much in the way of edible plants remaining. Most of the wild food plants in Europe tend to grow in hedgerows or the edges of woodland, and mushrooms will only grow in land that isn't frequently disturbed - so they'd have come from the meadows rather than the fallow land. And the wild food plants tend to be perennials rather than annuals, so wouldn't appear in the space of one fallow year. It's possible that in places where they weren't pasture that you could get a few nettles or the like, but probably not many.

Sources:

Warren Ault, Open-Field Farming in Medieval Europe: A Study of Village By-laws, Routledge, 2013

Mark Bailey, "Beyond the Midland field system: the determinants of common rights over the arable in medieval England" in Agricultural History Review, Volume 58, Number 2, December 2010, pp. 153-171

Thanks! Quality info in true r/AskHistorians style.