It is hard to diagnose without an image but it sounds like Cabbage Root Maggot, which is the larvae of a very small fly. It sounds as if you have an advanced infection and I suspect you do no want to use chemicals. So, there is very little that can be done, except working to prevent it in the future

Some resources from Canada (see also bottom of this answer, especially about it being an annual problem) http://www1.agric.gov.ab.ca/$departme...

This from UMass Amherst https://ag.umass.edu/vegetable/fact-s...
"Larvae feed on roots and can completely destroy the root system. The first sign of a problem is wilting of the plant on sunny days and yellowing or purpling of outer leaves. Later, plants collapse, wilt down, and die. On inspection of the root area you may find the legless white maggots feeding, the small brown, oblong pupae, or tunnels from maggot feeding. In brassica root crops such as turnips, radishes, and rutabaga, maggot feeding tunnels on or in the root render it unmarketable"

"Chemical Controls & Pesticides:
Direct application of insecticides to the root zone is considered the most effective means for controlling maggot damage. Insecticides should be applied as a narrow band with enough water to penetrate the root zone. For direct seeded crops, apply insecticides over the row. For transplanted crops, spray should be directed to the base of the plant. A wider spray band reduces the concentration of the insecticide over the row and therefore decreases its effectiveness. Some materials may be applied as a transplant tray drench or in transplant water, or in-furrow before or during seeding or transplanting. Be sure to read the label of any material you choose to use to determine which methods it is labeled for.
Scout each successive seeding or planting to determine whether treatment is required. Cabbage maggot eggs are very sensitive to high soil temperatures (above 95°F), and will die if they are exposed to these temperatures for several days in a row. Generally these soil temperatures are reached by late May/early June, unless there has been excessive rain, which has a dramatic cooling effect on the soil. This means that under high soil temperatures there is no need to spray for this pest.

Biological Controls:
Natural enemies. Soil-dwelling beetles, including ground beetles (carabids) and staphylinid beetles, feed on cabbage maggot eggs, larvae, and pupae and can cause high levels of mortality. One staphylinid species, Aleochara bilineata, also parasitizes maggot larvae and has been shown to respond to chemicals given off by plants that suffer maggot damage. Because these soil-inhabiting beetles are susceptible to insecticides, broadcast soil insecticide treatments should be avoided. Other natural enemies including parasitic wasps and predatory mites. Naturally-occurring fungal diseases occasionally will reduce onion maggot numbers, particularly when flies are abundant and relative humidity is high. During a fungal epidemic dead, diseased flies, can be seen clinging to the highest parts of plants along field edges.
Nematodes for biological control. Soil application of the entomopathogenic nematode Steinernema feltiae has shown efficacy against cabbage maggot in trials even at low soil temperatures 50°F/10°C). Apply by suspending infective juvenile nematodes in water and treating transplants prior to setting in the field (as a spray or soaking drench), or in transplant water used in the water wheel transplanter, as a drench after transplanting, or a combination of pre-plant and post-plant applications. Post-plant treatments are likely to be needed if maggot flight begins >1 week after transplanting. Rates of 100,000 to 125,000 infective juveniles per transplant have been shown to be needed to achieve reduction in damage. Nematodes need a moist soil environment to survive."

Canada Specific https://www.cog.ca/documents/RS14.pdf
"Cabbage root maggot
This insect attacks all brassicas, namely cabbage, cauliflower, broccoli, brussels sprouts, Chinese cabbage, kale, radish, turnips and rutabaga.
The adult is a fly, resembling a housefly, which lays its eggs on the soil near one of these plants. These longish white eggs can easily be noticed. In southern Ontario there are three generations of the fly emerging in May, early July and August. These are only two Generations starting
somewhat later in Quebec, the Atlantic Provinces and the Prairies, five Generations in coastal B.C.. starting much earlier. The first Generation is by far the most destructive. After hatching, the maggots enter the soil and feed on the rootlets of the stem brassicas (cabbage, etc.). Young plants become stunted and die. To prevent this, place a collar made of Tarpaper, with a hole in the middle and a slit, around the stem flat on the soil, so the fly cannot lay its eggs there.
Do this on the same day as transplanting. Instead of collars you can protect rows of plants by covering with Reemay cloth , which allows the passing of water, light and air. If you grow your own transplants, do the seeding in a frame covered with mosquito screen or
Reemay cloth and do the same for radish. Later generations do not kill established plants in mid summer. Winter turnips or rutabagas are riddled by the maggots but only superficially at the bottom of the root which can be trimmed of at harvest. By delaying the seeding until early to mid June this damage can be kept to a minimum. This insect is very common in the northern areas around the world and if present in your neighborhood it will be an annual problem"