Yes, you can extend most homes in North Finchley. Very little is listed. A handful of conservation areas cover particular streets. The clay underneath will matter more to your builder than the council does. The paperwork here is ordinary, and the foundations are the bigger job.
There are 26 listed buildings in and around North Finchley. None is grade I, two are grade II* and the other 24 are grade II. On a listed building, listed building consent is needed for almost any change to the fabric, inside as well as out. For a suburb of tens of thousands of homes, that's very few, so the chance of a typical semi or terrace being listed is small. Charles Jacques was building cottages on Lodge Lane in the 1820s, and the area has been adding rooms to its houses ever since with little to stop it.
Four conservation areas touch the North Finchley area: Finchley Church End, Finchley Garden Village, Glenhill Close and Moss Hall Crescent. Inside them, Barnet council takes a closer interest in what you do to the street-facing side of a house. That can bring extra consents for things that would go ahead without a form a few roads away, such as demolition, front extensions and some roof changes. Check Barnet's conservation-area map for the exact house, because the boundary can split a street and the house you're viewing may be on the stricter side of it.
People here alter their homes all the time. Barnet's planning records list 893 applications in the area between April 2024 and June 2026, and 337 in the last twelve months alone. There's a decent chance one of your neighbours has already done the extension you're thinking about, and their drawings are public.
The British Geological Survey puts North Finchley's ground at class 3 for shrink-swell, and all of the area reaches that class. Shrink-swell is clay that swells when it's wet and shrinks when it dries, so an extension built on it needs foundations deep enough to sit below the movement. Ask a structural engineer to look at the ground and any nearby trees before the design is fixed, because the foundation spec can change the cost more than the planning does.
Outside the conservation areas and away from the listed buildings, most houses in North Finchley can use the national permitted development rights for rear extensions, loft conversions and the like, within the usual size limits, with no full planning application. Flats work differently, because flats have no permitted development rights of this kind, and about half the homes here are flats. A flat owner's extension plans usually start with the freeholder.
For freedom to extend, North Finchley is middling. The council will mostly let a house owner get on with it, and the bigger hurdles are the clay underneath and, for flat owners, the freeholder.