Private landowners are not obliged to clear snow or ice from the pavement/highway, even if the road or pavement passes over their land since this is the responsibility of the local authority. From a legal point of view it may be risky for private individuals to clear these areas. By sweeping snow from one part of the pavement you can create a danger in another area and if someone is injured, you will be liable for negligence or nuisance.
On your own land, it is a different matter. You owe visitors (refer to legal definition in the Act) a duty under the Occupiers Liability Act 1984 to take reasonable care to ensure that they are reasonably safe. This means that if you know someone (such as the postman) is likely to walk up your garden path, and you also know that the garden path is slippery, you must take reasonable steps to clear the path of snow and grit it if necessary.
Best thing to do, would be to put a big warning sign up. The above duty would also apply if you have dodgy paving slabs or steps, or any trip hazards at all on your path.