When turf is wet and cold in winter it is very susceptible to damage - it can be destroyed in heavily used areas quite quickly.
But what about when the ground is frozen? Is the damage caused by movement of the mud/roots so that the stabilising effect of frozen ground makes it resistant to wear?
We keep people off our grass in the wet season but have a prolonged frozen period in the UK right now and the ground is well frozen... we can't decide if we can relax this rule.
I live in a cold and snowy region of the US. Here, if someone keeps walking over frozen lawn enough so that a path forms, then that grass will be much browner than normal once the snow melts. It will also be thinner when it greens up because the crowns were damaged by the foot traffic. The greater the traffic, the worse the damage. If a car parks on it, then the grass has a good chance of dying.
You might expect that snow will help prevent this, and to some extent it does, but snow underfoot quickly turns into ice, which is also not good for the grass crowns.
Be aware that, if I remember correctly, the type of grass in the lawn makes a difference as to how the lawn responds to the cold and foot traffic: a cool season grass handles the conditions better, as you'd expect, and fescue handles it better than bluegrass.
One important point—the lawn damage we can get here is the result of 6-10 weeks of walking several times a day on a path in the lawn. I suspect that this will not be the case in the UK (unless you live in the north of England or in Scotland?). You may, however, have to guard against damage caused by people walking over the lawn when the frost/ice in the soil is partially melted; this can cause uneven and rough surfaces that may require the area to be rolled or leveled in the spring.
I think the only danger from temporarily relaxing your rule about walking on grass when it is saturated is that people may think you've permanently gotten rid of the rule, thus making things harder for you and the lawn during the wet season.
You should avoid walking on a frosted or frozen lawn if possible - the blades of grass are frozen and therefore fragile even in a frost, and will be damaged by foot traffic. Once damaged, it may be more susceptible to disease in the lawn by spring/summer. https://www.uklawncare.net/lawn-maintenance/frost-on-lawns/
As with many things, it will be "dose dependent." Meaning, the more people walk on the grass the worse it will get. One person walking over probably won't make much difference. Thousands walking over will very likely heavily damage the grass.
In a connected thought, there is the "first bit of litter" rule. If there is no litter on the ground then people are reluctant to be the first person to litter. If there is already lots of litter around people hesitate less to add to it.
So keeping people off a bit of grass will leave it clear of foot prints. And people will be less likely to start making a trail. You might be able to keep them off with some little barrier at each end rather than putting up a huge fence. A 2 meter long bit of fencing up to knee high might be enough. They might choose to go somewhere else. Even a little "Please don't walk on the grass" sign might work. It depends where else they can walk.
Alternatively, if it is a place with unavoidable high traffic, you might have luck with providing a preferred path. Possibly you could provide something like patio stones or gravel or some such on the path. Or clear the snow and ice where you prefer they walk, to keep the damage in the shortest possible path.
