Greenway Lane, although not formalised as a postal address until the 1930s, pre-dates the 1773 map of Chippenham - indeed, Greenway Farm at the Bird's Marsh end of the road appears on that map, but the the road is known as "Greenaway Lane."
The road originally led to Greenways House, a former private residence that was eventually used as the local maternity unit until 1989. The name still lives on at Chippenham Hospital, but the house itself is long gone, replaced by a business park and an ambulance station.
Houses at the St Paul's Church end of the road can be seen on 19th century maps, as can South View - an older house close to the GZ - but the rest of the area wasn't built until well into the 20th century. Much of the land was formerly used for allotments and small scale agriculture, but was sold to developers in parcels - for example the sweep of Greenway Gardens was constructed in the late 1920s as a full elipse of houses, and only later were those houses now located on Greenway Lane absorbed into the street. Their numbers were changed to reflect their new location.
The original Chippenham Rugby Club ground was also located behind Greenway Lane, before moving to a location on the outer west side of town.
Edit: Congrats to GeoTigger for being FTF.