Local authorities have the power, under Section 27 of the Road Traffic Act 1988, to designate roads where it's an offence to permit a dog to be unless it's on a lead.
Under the Metropolitan Police Act 1839 (or the Town Police Clauses Act 1847, as appropriate depending on the location), it's an offence to allow a dog to put in fear any person or animal on the street. (In some cases, it could be argued that failing to keep a dog on a lead could contravene these laws).
The Dangerous Dogs Act 1991 makes it an offence 'allow a dog to be dangerously out of control in a public place'. (Once again, failure to keep a dog on a lead could, under some circumstances, fall foul of this legislation)
The Clean Neighbourhoods and Environment Act 2005 empowers local authorities to designate areas of land (including footpaths) where dogs must be kept on a lead.
The Control of Dogs Order 1992 requires all dogs, in a public place, to wear a collar which has the name & address of the owner on it (or on a tag attached to the collar). However, there's no requirement that the collar should be attached to a lead.
Otherwise, there's very little relevant legislation. The law places a duty of 'due care' on everybody. The courts could interpret this as meaning that dogs should be kept 'under proper control' in public places. A 'public place' doesn't have to mean publicly-owned land. It can also refer to any land to which the public has access, such as the type of lane referred to in the second part of your question.
Chris