For most homes here, yes, and with fewer obstacles than a village this old might suggest. The heritage footprint is light. Thirteen buildings carry a listing, and every one of them sits at the lowest grade. No conservation area covers the village itself. So unless you have bought one of a small, specific set of protected buildings, the rules on your extension are the ordinary ones rather than the restrictive ones.
The listed buildings are the exception that proves the point. All thirteen are grade two, the entry-level protection, with none at the higher grade two star and none at the top grade one. That still matters enormously if yours is one of them, because listing applies to the whole building inside and out, and it turns routine work into a consent job: the windows, the internal walls, the materials, all of it becomes a conversation with the conservation officer before a spanner moves. But thirteen buildings across a whole village is a short list. The overwhelming majority of homes here are not listed at all, and their owners never meet those constraints.
What clinches the freedom is what is absent. The village is not wrapped in a conservation area, the designation that quietly strips out permitted development rights and turns a new set of windows or a bit of cladding into a planning application. Without one sitting over your street, a typical house keeps its normal permitted development allowances, which means many extensions, loft conversions and outbuildings can proceed under those rights rather than needing full permission. That is the difference between a project you get on with and a project you negotiate. Two scheduled monuments sit in the wider area, but those are buried archaeology in the landscape rather than rules on your rear elevation, and they rarely touch an ordinary home improvement. The sensible step is still to confirm your specific property's status and check whether permitted development covers your exact plan before you commit, because the allowances have limits on size and position that a builder will know.
So can you extend a house in Alveston? For the ordinary, unlisted home on an unprotected street, comfortably yes, with more room to build than most villages of this age allow. Buy one of the thirteen listed buildings and the answer flips to a careful maybe with paperwork attached. Everyone else has one of the freer hands in the area.