Yes, for most houses in Enfield Highway, and heritage rules will rarely be what stops you. Listed buildings are few, and almost all of them are at the lowest grade. Only one scheduled monument is in reach. The neighbours file plenty of planning applications. What will slow an extension down is the clay underneath it.
Historic England lists 79 buildings in and around the area: two at grade I, two at grade II* and 75 at grade II. Listing covers the inside of a building as well as the outside, so any change that affects its character needs listed building consent on top of any planning permission, including work that would normally need no application at all. With 79 listed buildings spread over an area this size, almost every house a buyer looks at will not be one of them. The few that are come with a lot of paperwork.
The area has one scheduled monument, the strictest tier of protection there is, and it only matters to anyone planning work on or right next to it.
The planning register shows how often people here change their homes. The council logged 1,427 applications between April 2024 and June 2026, and 701 of them came in the last twelve months. That is a lot of loft conversions, rear extensions and new windows being proposed by people who live on the same streets. A new application on your road is more likely to be the house next door doing the same thing than anyone objecting to yours.
The ground needs more thought than the heritage does. On the British Geological Survey's shrink-swell measure, the area reaches class 3, and every postcode here is at that class or above. That means clay that swells when wet and shrinks when dry. Any extension needs foundations that go deep enough to allow for that movement, and the trees nearby make a difference too. Get a structural engineer to look at the ground early, since foundation depth can change the cost of an extension more than its floor area does. The new extension has to meet today's building regulations, on the same ground that has moved with the seasons since the original house was built.
One listed building in the area is on Historic England's Heritage at Risk register. That marks it as a building in poor condition. It may qualify for repair grants, and any work on it will be closely watched. For the buyer of an ordinary house down the road it makes no difference.
For a house that is not listed, a lot of modest extensions come under permitted development, which needs no planning application at all within national limits, and a lawful development certificate from the council confirms that a project qualifies. Look up the specific house on the council's planning map to see whether any local designation or condition removes those rights, because that is what decides whether an extension needs a full application or none.
Your freedom to extend here is middling: heritage protection hardly comes into it, and the real cost is digging foundations deep enough for the clay.