Your local authority is responsible for clearing snow and ice from the public highway and pavement. Under Section 41(1A) of the Highways Act 1980, the council must ensure that safe passage along a highway is not endangered by snow or ice. If the local authority fails to act, you can apply to the magistrates' court under the same legislation for an order to unblock the highway within a reasonable period - although such orders are very rare indeed.
Private landowners are not obliged to clear snow or ice from the highway, even if the road or pavement passes over their land. Indeed, 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 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.